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COPYRIGHT DEPOSIT. 



ARITHMETIC 

OF THE 

STEAM BOILEE 



THE POWER HANDBOOKS 

The best library for the engineer and the man who hopes 
to be one. 

This book is one of them. They are all good — and 
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SOLD SEPARATELY OR IN SETS 



By PROF. AUGUSTUS H. GILL 

OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY 

ENGINE ROOM CHEMISTRY 

By HUBERT E. COLLINS 

BOILERS KNOCKS AND KINKS 

SHAFT GOVERNORS PUMPS 

ERECTING WORK SHAFTING, PULLEYS AND 
PIPES AND PIPING BELTING 

By CHARLES J. MASON 
ARITHMETIC OF THE STEAM BOILER 



McGRAW-HILL BOOK COMPANY, Inc. 

239* WEST 39TH STREET, NEW YORK 

6 BOUVERIE STREET, LONDON, E. C. 



THE POWER HANDBOOKS 

5 

ARITHMETIC 

OF THE 

STEAM BOILER 

A REFERENCE BOOK 

SHOWING THE VARIOUS APPLICATIONS OF 

ARITHMETIC TO STEAM BOILERS 



BY 

CHARLES J. MASON 



First Edition 



McGRAW-HILL BOOK COMPANY, Inc. 
239 WEST 39TH STREET, NEW YORK 

6 BOUVERIE STREET, LONDON, E. C. 

1914 



TJ ■X16 

.Hi. 



Copyright 1913, by the 
McGraw-Hill Book Company, Inc. 









THE.MAPLE.PRESS.YORK-PA 

JAN -9 1914 

©CI.A361529 









c 



THIS BOOK 
IS RESPECTFULLY DEDICATED 

TO 

MR. FRED R. LOW, EDITOR OF POWER, 

WHO HAS EVER TAKEN A DEEP INTEREST IN THE 

AFFAIRS OF THE ENGINEERING FRATERNITY 



PREFACE 

This book is a compilation of arithmetical rules and 
formulas applicable to steam boilers of various types. 
The author claims no originality in the preparation of 
the material, excepting only the arrangement and manner 
of presentation. 

It is intended as a book of reference for those who 
may require rules and formulas directly related to steam 
boilers, and its aim is concentration and logical order in 
the arrangement and treatment of the various features 
introduced. 

Most of the material was gathered during the author's 
career as a steam and marine engineer, covering a period 
of twenty-five years. 

It is not intended to teach the elements and principles 
of arithmetic in this book, as might perhaps be inferred 
from the title, but only the application of arithmetic to 
steam-boiler calculations. It is presumed that those 
who may use it already understand arithmetic but 
desire to have a compact set of rules and formulas con- 
veniently ready for use, without having to look through 
several books for a certain one when required. Those 
who are preparing for examinations for engineer's certifi- 
cates and licenses will find the work of great assistance 
to them. 

The author desires to thank all those who have in 
any way contributed to the production of this work, 

vii 



viii PREFACE 

particularly the publishers and editor of Power, from 
which paper various extracts have been taken, kind 
permission to use the same having been granted, and 
Mr. William Kent, M. E., author of Kent's Mechanical 
Engineer's Pocket-book. 

Charles J. Mason. 

SCRANTON, PENNA, 

December, 1913. 



CONTENTS 

Page 
Preface vii 

PART I 

CHAPTER I 

The Sphere, Stress per Square Inch Section; Safe 

Pressure i 

Calculations pertaining to the sphere — the cylinder — 
riveted joints — efficiencies — bursting and safe pressures. 

CHAPTER II 

Boiler Heads. — Unstayed Heads 30 

Boiler heads — unstayed bolts — convex and concave — flat 
unstayed heads — stays and staybolts — diagonal stays — 
segments to be braced — girder bars. 

CHAPTER III 

Manhole Reinforcing Rings 56 

Reinforcing rings — heating and grate surface — corrugated 
furnaces — horse-power of boilers — ratio of heating to grate 
surface — equivalent evaporation — boiler efficiency — boiler 
trials. 

PART II 

Miscellaneous .Applications 83 

Bursting pressure of pipe — cost of evaporating 1000 lb. of 
water — safe pressure of flat cast-iron heads — equivalent 
boiler performance — efficiency of diagonal seam — collap- 
sing strength of cone-shaped flue — strength of cone seam — 
safety valves — Roper's rules — tapered levers — chimneys 
— size of feed pipes 

ix 



x CONTENTS 

PART III 

Page 

Appendix 133 

Abstracts from rules, United States Board of Supervising 
Inspectors of Steam Vessels — abstracts from Massachu- 
setts' Boiler Rules. 

Tables 191 

Diameters, areas and circumferences of circles — decimal 
equivalents — squares, cubes, cube roots and square roots 
— factors of evaporation — standard boiler tubes — Kent's 
table of chimneys — Mark's and Davis' steam tables. 

Index 221 



ILLUSTRATIONS 

Figure Page 

i Diagram of stresses in a sphere 2 

2 The cylinder 6 

3 Types of riveted joints 12 

4 Data sheet, double butt-strapped joint 22 

5 Quadruple, double butt-strapped joint 25 

6 Bumped heads 30 

7 Diagram to find radius of a bumped head 31 

8 Arrangement of direct and diagonal stays 37 

9 Diagonal stays 40 

10 Segment of head to be braced 48 

11 Approximate method of finding area of segment 51 

1 2 Girder bars 53 

13 Direction of stress in reinforcing rings 60 

14 Lower part of Manning boiler 86 

15 Diagonal steam 91 

16 Cone-shaped flue ! 93 

17 Strength of seam in cone 94 

18 Diagram of fire box 97 

19 Diagram of safety valve dimensions for calculations 100 

20 Diagram showing tapered safety valve lever for calculation. 1 13 

21 Diagram of locomotive boiler 130 



XI 



PART I 

BOILER CALCULATIONS 



BOILER CALCULATIONS 

CHAPTER I 

The Sphere; Stress Per Square Inch Section; 
Safe Pressure 

The sphere is the strongest form in which a steam 
boiler could be made, but because of mechanical and 
commercial reasons, that form is not used. In order to 
understand the stresses, due to pressure, endured by 
steam boilers, it is well to start with the spherical form, 
for that is the simplest, and it forms a basis for calcula- 
tions on the prevailing forms in which boilers are made. 

Given a spherical vessel made of metal of a certain 
thickness and known diameter, it is desired to find what 
stress per square inch section the metal is subjected to, 
due to a known pressure per square inch contained within 
the sphere. The pressure would tend to separate the 
sphere in halves, through a diametral plane. Actually, 
pressure in a closed vessel of any form radiates from the 
center outward. But for convenience in calculations 
the radiating forces may be resolved in two, and acting 
perpendicular to any diametral plane. The total pres- 
sure or force tending to burst the sphere asunder will be 
the product of the known pressure per square inch, 



2 ARITHMETIC OF THE STEAM BOILER 

existing in the sphere, and the area in square inches of 
a diametral plane. 

For illustration, assume a pressure of ioo lb. per square 
inch; a sphere whose internal diameter is 30 in., and made 




Diametrical Plane 

Arrows show Direction of Resolved Forces 



Shaded Rings shows Thickness of Sphere 



Hemisphere 

Fig. i.— Diagram of stresses in a sphere. 

of 1/2-in. steel plate, no joints, seams, nor rivets. 
The area of the diametral plane is: 

30 2 X. 7854 = 706.860 sq. in. 
As the pressure is 100 lb. on each square inch, then: 
100X706.860 = 70686.00 lb. 

This is the total force tending to separate the sphere in 

two parts. 

This force is resisted by the area of metal at the cir- 



THE SPHERE 3 

cumference of the plane upon which the pressure is as- 
sumed to act. This is actually a ring whose internal 
diameter is 30 in. and whose outside diameter is 31 in. 
This is called sectional area, and it is the difference be- 
tween the area of the inner and outer circles of which 
the ring of metal is formed. 

The area of a circle whose diameter is 31 in. is: 

3 i 2 X. 7854 = 754.769 sq. in. 

and, 30 2 X. 7854 = 706.860 sq. in. 

Difference = 47.909 sq. in. 

That is, 47.909 sq. in. cross-sectional area of steel plate 
is enduring a total stress of 70,686 lb. or, 

70686 



47.909 



= 1475.42 lb. 



per square inch section. 

Assuming the tensile strength of the plate to be 60,000 
lb. per square inch section, the foregoing shows that 
considerably more than 100 lb. per square inch could 
safely be carried in the sphere considered, for with a 
factor of safety of 6, the stress would be 10,000 lb. and 
this would be obtained by having the pressure at 677.77 
lb. per square inch, as a trial in calculation will show. 

Ordinarily, in practical work, the difference between 
the outside and inside diameters is not taken. 

For example, the rule that would be employed to find the stress 
per square inch in a sphere whose inside diameter is 30 in., and 
whose thickness of plate is .5 in., is: Multiply the area of the dia- 
metral plane by the pressure per square inch, and divide that 
product by the product of the inside diameter, the constant 3. 141 6, 



4 ARITHMETIC OF THE STEAM BOILER 

and the thickness of the plate in inches. This written in formula 
is: 

d*X-7SS4XP . 

d X3.i4i6X*~ StreSS - 

But by simple cancellation the formula reduces to 

pXd 

—ttt = stress. 

4Xt 

The value of the letters used is: 

d= inside diameter, inches; 
p= pressure, pounds per square inch; 
t= thickness of plate, inches; 
s= safe stress in pounds per square inch; 
.7854= a constant; 
3.1416= a constant. 

In order to find the pressure per square inch that may 
safely be carried, when the safe stress, the diameter of 
sphere, and the thickness of plate are given, it is simply 
a matter of changing the rule and formula to suit the 
purpose, thus: 

Multiply four times the thickness of plate by the given 
safe stress per square inch section, and divide by the 
internal diameter in inches. 

Written in formula it is : 

4/ X stress 

-j--=p. 

Applying this to the example chosen, it becomes: 
4X .5X10000 



3° 



= 666.66 lb. 



THE SPHERE 5 

which for practical purposes is 667 lb. per square inch. 
Here, the internal diameter only has been taken. In the 
first method shown the mean diameter was taken, which 
gave a safe pressure of 677.77 lb. per square inch which 
for practical purposes is 678 lb. The difference is 11.1 
lb. or 1.6+ per cent, difference in favor of the usual 
method, which, though not absolutely correct, errs on the 
side of safety as shown. 

If the sphere in the example were .25 in. thick, or 60 in. 
internal diameter, then the difference in the methods 
explained would be less than that given. 

Grouping all the rules and formulas pertaining to the 
sphere, so that any term may be found having the remain- 
ing terms given: 

To find the stress per square inch section endured by the plates, 
multiply together the pressure per square inch and the internal 
diameter in inches. Divide the product by four times the thickness 
of the plate in inches. 

Written as a formula this rule becomes: 

pXd 

4X7 = * (I) 

To find the safe pressure per square inch that may be carried, 
multiply four times the thickness of the plate by one-sixth of the 
ultimate tensile strength of the material of which the plates are 
made, and divide the product by the internal diameter in inches. 

4XtXs 

To find the internal diameter, multiply four times the thickness 
of the plate by one-sixth of the ultimate tensile strength of the 



6 ARITHMETIC OF THE STEAM BOILER 

material of which the plates are made, and divide by the safe pres- 
sure per square inch. 

^ = <*. (3) 

P 

To find the thickness of plate, multiply the internal diameter by 
the safe pressure per square inch, and divide the product by one- 
sixth the ultimate tensile strength of the material of which the plates 
are made; divide the quotient by 4. 

JXs~ L (4) 

The Cylinder 

Next to the sphere, the cylinder is the form best suited 
for steam boilers, and excepting the spherical form, the 




(a) ' W 

Fig. 2. — Stresses in cylinders, (a) Stress acting at right angles 
to the longitudinal plane abed. Tendency to separate the cylin- 
der at a d and b c. (b) Stress acting parallel to the longitudinal 
plane abed. Tendency to separate the cylinder at ef. 

cylindrical is the strongest. This is not hard to under- 
stand, when it is considered that pressure inside any 
vessel tends to make that vessel assume a spherical form, 
as illustrated in the inflated toy rubber balloon. For 
this reason flat surfaces in boilers must be braced, but 
cylindrical surfaces do not require any bracing. 



THE CYLINDER 7 

For the sake of convenience in calculations, the force 
due to the pressure in a cylindrical vessel may be consid- 
ered as acting in two directions. One acts in the direc- 
tion tending to blow off the head, and the other at 
right angles to the diametral longitudinal plane. In the 
former, the calculations are exactly the same as has been 
explained in connection with the sphere. In the latter, 
the pressure acting against the imaginary plane tends to 
separate the cylinder into two parts, and that which 
resists the tendency is the area of metal along both sides 
of the cylinder. 

The stress per square inch section of the plate, due 
to any given pressure, may be found from the follow- 
ing rule: 

Multiply together the pressure per square inch, the diameter of 
the cylinder in inches, and the length in inches; divide the product 
by two times the thickness of the plate in inches multiplied by the 
length in inches. 

This, written as a formula, is: 
pXdXl 



2XtXl 



(S) 



As the factor / appears both above and below the line, 
it may be cancelled, and the formula becomes, 

From the formula just given it can be seen that the 
stress increases as the diameter or pressure increases, and 
it also can be seen that as the thickness of the plate 
increases the stress on the plates decreases. 

By comparing formula (1) with formula (6) it is seen 



8 ARITHMETIC OF THE STEAM BOILER 

that in the latter the stress is twice as great as it is in the 
former. A careful study of these rules and formulas 
will show why the stress in one case is just twice that of 
the other. For that reason, the longitudinal seams in a. 
boiler must be made stronger than the circumferential 
seams, and this is accomplished by having more than 
one row of rivets in the longitudinal seams. 

Circumferential seams are frequently double riveted 
to make a mechanically tight job. The strength of riv- 
eted joints will be treated further on. So far, seamless 
vessels have been considered, so as to introduce the sub- 
ject of stress and resistance to stress in the most simple 
form. 

The following rules relate to the cylindrical form, and 
belong in the group under consideration. 

To find the stress on each inch in the circumference 
tending to blow off the head in a longitudinal direction 
(not the stress per square inch section), the following rule 
is applicable : 

Multiply together the area in square inches of the end of the 
cylinder and the pressure per square inch, and divide the product 
by the circumference of the cylinder in inches. 

<* 2 X. 7854x1 

JX3.1416 ""*■ {7) 

To find the total stress caused by the pressure in a cylinder, 
multiply together the diameter in inches, the length in inches, 
and the pressure in pounds per square inch. 

d XI Xp = totals (8) 

/ = length in inches. 

To find the total pressure on the entire shell of a cylinder, multi- 



THE CYLINDER 9 

ply together the circumference in inches, the length in inches, and 
the pressure per square inch. 

cXlXp = total pressure. (9) 

To find the bursting pressure of a cylinder, multiply together 
the thickness of the plate in inches and the tensile strength of the 
material of which the plate is made, and divide the product by 
the radius of the cylinder in inches. 

ZX T 

= bursting pressure per square inch. (ro) 

In this formula, t = thickness of plate in inches. T = tensile 
strength in pounds per square inch section of the material, and 
r = the radius in inches. 

To find the safe working pressure of a cylinder, multiply together 
the thickness in inches and the tensile strength of the material of 
which the plate is made, and divide that product by the radius 
in inches multiplied by whatever factor of safety may be desired. 

tX T 

,. = safe working pressure per square inch. (n) 

rX/ 

Here follows examples showing the application of the 
foregoing rules and formulas from (1) to (n) inclusive. 
For the sake of clearness and convenience the same values 
will be used in all. Numbers easy to operate have been 
chosen, for no matter what numbers may be contained 
in any example which may come up in practice, the 
method of operation will be exactly the same. 
In formula (1) assume the following values: 

pressure (p) =100 lb. per square inch; 

diameter (d) = 60 in. ; 

thickness (t) =1/2 or .5 in. 

Then to find the stress (s) the statement becomes: 

100X60 . 

— — = 3000 lb. per square inch section. 

4X -5 



10 ARITHMETIC OF THE STEAM BOILER 

To find the safe pressure (p) which may be carried, 
formula (2), the statement becomes: 

4X -5X3000 „ . . 
= 100 lb. per square inch. 

To find the internal diameter (d), formula (3), the 
statement becomes: 

4X .5X300 , . 

- = 60 in. 
100 

To find the thickness of plate (/), formula (4), the 
statement becomes: 

100X60 

= .5 in. 

4X3000 

Formula (5) reduces to that given in (6), and to find 
the stress per square inch section in this case, the state- 
ment becomes: 

100X60 . n 

— — = 6000 lb. 

2X.5 

per square inch section. 

To find the stress (s) on each inch in the circumference, 
formula (7), the statement becomes: 

6o 2 X. 7854X100 

— z~r^ 7— = 1500 lb. 

60X3. 1416 J 

To find the total stress on the entire shell, formula (8), 
assume a length of 144 in. with the other values remaining 
the same, the statement becomes: 

60X144X100 = 864,000 lb. 



THE CYLINDER II 

To find the total pressure on the entire shell, formula (9), 
the statement becomes: 

60X3.1416X144X100=2,714,342.4 lb. 

To find the bursting pressure, formula (10), the state- 
ment becomes: assume a tensile strength of 50,000 lb. 
per square inch section. 

.5X50000 n . ,. . . 

— — = 833.33+ lb. per square inch. 

To find the safe working pressure, formula (n), and 
assume a factor of safety of 5, the statement becomes: 

s X ^0000 

^^b = 166.66+ lb. 

30X5 

In practice 170 lb. would be allowed with factor of 
safety 5 per square inch. Or a simpler method is to 
take one-fifth of the bursting pressure thus: 

83 ^=i66.66 1b. 
5 

and as the bursting pressure is five times the safe pressure, 

5X166. 66+ = 8 33 . 33+ lb. 

In actual practice, 6 is used as a factor of safety more 
frequently than 5. However, the method of operation is 
the same for any and all factors of safety that may be 
used; so that if the method is understood, it matters not 
as to the values that may be substituted in the various 
formulas treated of. 



12 



ARITHMETIC OF THE STEAM BOILER 



Riveted Joints 

In the previous sections, spheres and cylinders without 
joints or seams were assumed in order to simplify explana- 



Pitch 





Pitch 



Pitch 




A 



© 






©!© © © ©^ 
>©©©©_© 
© © © © 
©i © © © ©!© 



~^- 



w 



Fig. 3. — Types of riveted joints. (A) Single-riveted lap joint; 
one rivet in single shear. (B) Double-riveted lap joint; two rivets 
in single shear. (C) Triple- riveted, double butt-strapped joint; 
four rivets in double shear, one rivet in single shear. (D) Quad- 
ruple-riveted, double butt-strapped joint; eight rivets in double 
shear, three in single shear. 

tions and calculations. In actual practice, however, 
steam boilers are constructed with both longitudinal 



RIVETED JOINTS 13 

and circumferential joints, or seams. The seams are 
secured by rivets. There are several kinds of riveted 
joints known to engineers and others who have to do 
with boilers. The strength of a riveted joint depends 
upon how the joint is made, as to the size and pitch or 
spacing of the rivets, and the number of rows of rivets, 
and also as to whether the joint is what is known as a lap 
or butt-strapped one. A riveted joint of any kind is not 
theoretically as strong as the solid part of the plate, 
although in practice it has been known for boilers to 
tear apart at some place other than at the riveted joint. 
This probably was due to a flaw or weakness in the metal. 
If a cylindrical vessel were made of plates uniform in 
structure and thickness throughout, and if tested to 
destruction, it would likely break or pull apart at the 
riveted longitudinal joint. In making calculations it is 
presumed that a break would occur at the joint, rather 
than at any other place. 

Strength of Riveted Joints 

The strength of a riveted joint is compared with that of 
the solid plate, the latter being valued at 100 per cent. 
Nominally, the strength of joints varies from 56 per 
cent, to 94 per cent., the former value representing single- 
riveted lap joints, and the latter, quadruple, double 
butt-strapped joints. Between single-riveted lap joints 
and quadruple-riveted, double butt-strapped joints, 
there are: double-riveted lap joints, triple-riveted lap 
joints, quadruple-riveted lap joints, single-riveted butt 
joints, double-riveted butt joints, triple-riveted butt 
joints. The foregoing joints may be either chain riveted, 



14 ARITHMETIC OF THE STEAM BOILER 

or what is termed zig-zag riveted. In triple-riveted 
joints and in quadruple-riveted joints it is customary to 
omit every alternate rivet in the outer rows; this admits 
of a stronger joint. 

The strength of a riveted joint depends upon the size 
and pitch of the rivets, the number of rows of rivets, 
type of joint as to lap or butt, the tensile strength of the 
plates, and the shearing stress of the material of which 
the rivets are made. In rinding the strength of a joint 
two things are to be considered, the strength of the plate 
section and the strength of the rivet section. The 
lesser value, as found from an analysis, is taken as the 
strength of the joint as a w T hole. 

Theoretically, riveted lap joints and those butt joints 
with one cover plate should be designed so that the rivet 
and plate sections are equal — or as nearly equal as possi- 
ble — in strength. But in practice it is usually considered 
desirable to so design a joint that the plate section is a 
little stronger than the rivet section; this particularly 
relates to joints having all the rivets either in single or 
double shear, for the reason that the plates become thin 
from w T ear, with a consequent reduction in strength, 
while the rivets suffer little if any from wear. But in 
joints having some of the rivets in single shear and some 
in double shear, the greatest strength usually obtains 
when the rivet section exceeds the strength of the net 
section of plate. 

Efficiency of Riveted Joints 

To determine the efficiency of a riveted joint, its re- 
sistance must be calculated for each of the different ways 



RIVETED JOINTS 15 

in which it may fail, and then the lowest efficiency so 
found in relation to the solid plate, will be the one by 
which the joint is known. A riveted joint may fail in the 
following ways: 

(1) The plate may break asunder along the rivet holes, 
at the net section. 

(2) The rivets may shear off, leaving the plates intact. 

(3) The plate may shear out in front of the rivets. 

(4) The plate may crush in front of the rivets. 

(5) In joints having zig-zag rivets the plate may break 
diagonally between the rivet holes. 

(6) The joint may fail by a combination of the fore- 
going. With joints as usually proportioned, the liability 
to failure in the ways referred to in (3), (4), (5) and (6) 
is reduced to a minimum; this by having the distance from 
the edge of the plates to center of rivet holes one and one- 
half times the diameter of the rivet holes. It is custom- 
ary (except in special cases which will be referred to) to 
consider only (1) and (2) as possible ways of failure, and 
base all calculations upon those two ways. Therefore, 
calculate the efficiency of the net section of the plate 
part of the joint as compared with the solid plate, and 
then find the efficiency of the rivet section of the joint as 
compared with the solid plate; the lesser of the values 
found is to be taken as the final efficiency of the joint as a 
whole. 

Single-riveted Lap Joints 

To find the efficiency of a single-riveted lap joint. 
(The distance from the edge of the plate to the center 



1 6 ARITHMETIC OF THE STEAM BOILER 

line of rivet holes must be not less than one and one-half 
times the diameter of the rivet hole, for all joints.) 

First, find the strength of a unit of length of the solid 
plate. 

PXtX S = strength of solid plate. 

In which, 

P = pitch of rivets in inches, from center to center. 

/ = thickness of plate in inches. 

5 = tensile strength of plate, in pounds per square inch 

section. 
d = diameter in inches of rivet holes. 

The next step is to find the strength of the net section 
of plate between the rivet holes: 

(P—d)XtXS = strength of plate between the holes. 

Next, find the shearing strength of one rivet in single 
shear: 

nXsXa = shearing strength of one rivet. 

n = number of rivets in single shear; 5 = shearing strength 
of rivet; a = cross-section area of rivet, after driving. 

Take the lesser of these two results and divide it by 
the value found for the strength of the solid strip; 
the quotient will be the efficiency of the joint decimally 
expressed. 

Example. — Single-riveted lap joint of the following dimensions: 
•S =55,ooo lb. 
P =1.625 in. 
t =.25 in. 
d = .6875 in. 
5 =42,000 lb. 
a = .3712 sq. in. 



RIVETED JOINTS 1 7 

The strength of the solid strip will be: 

i. 625 X. 25X55*000 = 22,343 lb. 

The strength of the net section of plate between the rivet holes 
will be: 

(i. 625-. 6875) X. 25X55,000 = 12,890 lb. 

The strength of the rivet in single shear will be: 

1 X42,oooX . 3712 = 15,590 lb. 

As the net section of plate in this example is weaker than the 
rivet, its value must be used. Then: 

12800 . 
= .576, or 57.6 per cent, efficiency. 

Shearing Strength or Rivets 

The shearing strength of rivets may be taken from the 
following table (from the Massachusetts Board of Boiler 
Rules) : 

Iron rivets in single shear, 38,000 lb. 

Iron rivets in double shear, 70,000 lb. 

Steel rivets in single shear, 42,000 lb. 

Steel rivets in double shear, 78,000 lb. 

These values are on the safe side, as they are lower 
than some others that are in use. 

Double-riveted Lap Joints 

To find the efficiency of double-riveted lap joints, the 
method of procedure is the same as that for single- 
riveted joints, with the exception that there are two rivets 
in single shear instead of one as in single-riveted joints. 



18 ARITHMETIC OF THE STEAM BOILER 

Example. — A double-riveted lap joint has the following 
dimensions: 

5 =55,000 lb. 

t = .3125 (5/16) in. 

P =2.875 (2 7/8) in. 

d = -75 (3/4) in. 

a = .4418 sq. in. 

5 =42,000 lb. 
The strength of the solid plate is: 

2. 875X. 3125X55,000=49,4*4 lb. 

The strength of the net section of plate is: 

(2. 875-. 75) X. 3125X55,000 = 36 ? 5 2 3 lb. 

The strength of the two rivets in single shear is : 

2X42, 000X .4418 = 37,111 lb. 

Here again the plate section is the weaker, so the value for that 
must be used: 

— = . 739, or 73 . 9 per cent, efficiency of joint. 
49414 

Triple- and Quadruple-riveted Lap Joints 

In triple-riveted and quadruple-riveted lap joints 
(sometimes used in marine boilers) there are three and 
four rivets, respectively, in single shear. With this 
exception, the method of finding the efficiency of such 
joints is the same as for single and double, as just 
illustrated. 

Lap Joints 

Lap joints for the longitudinal seams are now consid- 
ered not safe for steam boilers of more than 36 in. in 



RIVETED JOINTS 



19 



diameter, and for pressures higher than 100 lb. per square 
inch; and probably as time goes on they will not be used 
at all; but it is important to know how to calculate the 
strength of such joints, hence the reference to them in 
this book. 

Butt Joints 

Butt joints with double cover plates are the strongest 
and safest joints in use. The minimum thickness of 
cover plates, or butt straps as otherwise called, is as follows : 

PRESCRIBED BY THE MASSACHUSETTS BOARD OF 
BOILER RULES 



Thickness of shell 


Minimum thickness of 


plates 


butt straps 


1/4 in. 


1/4 in. 


5/16 in. 


1/4 in. 


3/8 in. 


5/16 in. 


7/16 in. 


3/8 in. 


1/2 in. 


7/16 in. 


9/16 in. 


7/16 in. 


5/8 in. 


1/2 in. 


3/4 in. 


1/2 in. 


7/8 in. 


5/8 in. 


1 in. 


3/4 in. 


1 1/8 in. 


3/4 in. 


1 1/4 in. 


7/8 in. 



Single Butt Straps 

Single butt straps should never be thinner than the 
plates of the shell. In some instances (British Board of 
Trade and Canadian Rules) the minimum thickness 
must be not less than one and one-eighth the thickness of 
the shell plates. Double butt straps must be at least 

3 



20 ARITHMETIC OF THE STEAM BOILER 

five-eighths, and preferably the thickness of the shell 
plates. If the shell plate is light, say 7/16 in. or less, 
the outside strap should be as heavy as the plate, to 
admit of a tightly calked joint. 

When single butt straps are used, the method of 
finding the efficiency of the joint is the same as that 
for lap joints, for the rivets are all in single shear, and 
the pitch of the rivets is the same in each row. 

Number of Rivets Considered 

In lap joints, all the rivets in a given pitch strip of 
plate are taken into account when figuring for the effi- 
ciency of joint, while in butt joints only those rivets on 
one side of the center line of the joint are considered. A 
little thought on the part of the reader will make clear 
the reason. 

Number oe Rows of Rivets 

In double butt-strapped joints, three or four rows of 
rivets on each side of the center line are generally used. 
In the former (triple riveted) the pitch of the outer row 
of rivets on each side of the center line is twice the pitch 
distance of the two inner rows of rivets on each side of 
the center line. 

In the latter (quadruple-riveted joints) the outer row 
of rivets on each side of the center line of joint is four 
times the pitch distance of the two inner rows on each 
side of the center line; in the rows next to the outer rows, 
the rivets are pitched twice the distance of those in the two 
inner rows. (See Fig. 3.) 



RIVETED JOINTS 21 

Number of Rivets in Double and Single Shear 

In triple-riveted butt joints, there are four rivets in 
double shear, and one rivet in single shear, in a given 
pitch strip. 

In quadruple-riveted butt joints, there are eight rivets 
in double shear, and three rivets in single shear, in a given 
pitch strip. 

When calculating the efficiency of triple- and quadruple- 
riveted joints, the strength of the net section of plate is 
taken at the outer row of rivets, where the pitch is the 
greatest. The reason for this will be explained presently. 

High Joint Efficiencies Due to Wide Spacing of 
Rivets at the Outer Rows 

It is because of the wide spacing in the outer rows of 
rivets that such high efficiencies can be obtained with 
those types of joints as compared with those joints in 
which the pitch of the rivets is the same for all the rows. 

In order to explain why the net section of the plate at 
the outer row of rivets is taken, an illustrative example of 
a triple-riveted, double butt-strapped joint will be used; 
the same principles may be applied to a quadruple joint 
of the same kind. 

Thickness of shell plates 3/8 in., tensile strength 
50,000 lb. per square inch section, rivet holes 13/16 in., 
rivets 3/4 in. diameter; shearing stress of the rivets 
taken as 38,000 lb. per square inch section. The pitch 
of rivets in the two inner rows is 3 1/4 in., and in the outer 
row, 61/2 in. 



22 



ARITHMETIC OF THE STEAM BOILER 



The width of strip to be considered in this case is 6 1/2 
in. The sectional area is .375X6.5 = 2.4375 sq. in. 
2.4375X50,000=121,875 lb. strength of the solid strip, 
with which the joint is to be compared. 

Next find the strength of the net section of the plate 
at the outer row of rivets. 

As there is but 1 rivet in the 6 1/2 in. strip under con- 




Inside 
Cover Plate 

Outside 
Cover Plate 



Bottom 

Fig. 4. — Diagram of triple-riveted double butt-strapped joint. 



sideration, at the outer row, then, 6. 5— .8125 = 5.6875 
in. width of net section of plate; and, 5. 6875 X. 375 X 
50,000=106,640 lb. strength of net section of plate 
between the rivet holes. 

If the plate should break or pull asunder at the line 
of the outer row of rivets, the resistance to breaking is 
the metal in the net section of the plate as shown at c 
in Fig. 4. But should the plate break along the net 



RIVETED JOINTS 23 

section at the inner row of rivets, the resistance offered 
is the strength of the sections of plate E D and F, and 
as well as that, the resistance to shearing offered by 
one-half of each of the rivets in the outer row, which is 
equivalent to one rivet in calculation. 

First, find what the plate resistance is. By measure- 
ment it is 6.5 — (2X .8i25)=4.875 in. Here there are 
two rivet holes to subtract from the width of the strip. 
The sectional area will be 4. 875 X .375 = 1 .828 sq. in.; 
and 1.828X50,000 = 91,400 lb. resistance. 

To this must be added the resistance offered by the 
two half rivets in the outer row. The area of a 13/16-in. 
driven rivet is .5185 sq. in., and as the shearing stress is 
38,000 lb. per square inch, .5185X38,000=19,703 lb. 
This added to 91,400 lb. before obtained for the plate 
gives 111,103 lb. total resistance to the plate breaking 
at the inner row of rivets, as against the 106,640 lb. 
found for the net section of plate at the outer row of 
rivets. Therefore the latter is the weaker, and there 
the plate w T ill probably break, if at all. 

Continuing, there are four rivets in double shear, and 
one rivet in single shear, in the strip. The resistance to 
shearing of one rivet was shown to be 19,703 lb. The 
resistance to shearing offered by each rivet in double 
shear is: 

38,oooX .85=32,300 and 38,000+32,300 = 70,300 lb. 
per square inch section. 

(The factor .85 is a value used for rivets in double 
shear as double shear does not necessarily mean twice 
that of single shear.) The area of each rivet is .5185 



24 ARITHMETIC OF THE STEAM BOILER 

sq. in., and there are four to consider, therefore, .5185 
X4X7o,3oo= 145,802 lb. and to this add the value for 
the rivet in single shear, 19,703 lb., which gives a total of 
165,505 lb. shearing strength of all the rivets in the 
strip. The net section of plate at the outer row of rivets 
is the weakest part of the joint as a w T hole, and its value 
is to be compared w T ith the strength of the solid strip in 
order to find the efficiency. The net section of plate is 
106,640.625 lb. and the solid strip is 121,875 lb.;theeffi- 
ciency is: 

106640.62=; 

—^ = 87. s per cent. 

121875 ' J ^ 

Quadruple-riveted, Double Butt-strapped Joint 

To analyze a quadruple-riveted joint and find the 
efficiency, proceed as follows: 

Fig. 5 shows the construction and arrangement of 
rivets. The data is given in this manner. A strip of 
the joint marked P is taken. The value of the letters is: 
P = pitch of rivets in inches. 
/ = thickness of plate in inches. 
S = tensile strength of plates. 
d = diameter of the driven rivets, in inches. 
N = number of rivets in double shear. 
n = number of rivets in single shear. 
a = area of cross-section of rivets, in square inches. 

Strength of the solid strip of plate considered = 
PXtXS, represented by letter A. 

Strength of plate between the rivet holes at the outer 
row of rivets = (P — d)X S, represented by the letter B. 



RIVETED JOINTS 



25 



The shearing strength of 8 rivets in double shear, plus 
the shearing strength of 3 rivets in single shear = Na+na, 
represented by the letter C. 

The strength of the plate between the rivet holes in the 
second row plus the shearing strength of 1 rivet in single 
shear in the outer row = (P — 2d)XtX S+ na, represented 
by the letter D. 

Next, divide B, C, or D, whichever is the least in value, 



<■ 



4> 



±V 



Q 






O 



O- 



$- 



e 



o 



e 



r — 
^ — 



tx> 



e- 



-e- 



■& 



e 



0- 



^? 



f 

1 L 



e 






^> 



& 



-e 



O-O 



e 




-0- 



e- 



e 



^ 



e- 



_ P — 









Fig. 5. — Quadruple-riveted double butt-strapped joint. 



by the value of A, and the quotient w T ill be the efficiency 
of the joint. 

The numerical values to be used are: 
5 =55,000 lb. 
/ = 1/2 in. or .5. 
P =15 in. 

d =15/16 in. or .9375 in. 
a =.6903 sq. in. 

N =8 rivets, double shear value of 78,000 lb. 
n =3 rivets, single shear value of 42,000 lb. 



26 ARITHMETIC OF THE STEAM BOILER 

The final values are: 
A =isX. 5X55,000 = 412,500. 
B =(i5-.9375)X. 5X55,000 = 386,718. 
C = 8X78,oooX. 6903+3 X4-2,oooX. 6903 = 517,725. 
D =(i5-2X.937S)X.5X55 J oo°+ I X 42,000 X .6903 = 
389,930. 

The value of B is found to be the least, therefore the 
strength of the joint depends upon the weakest part, 
and the efficiency is, 

386718 



412500 



.937 or 93. 7 per cent. 



The foregoing example shows how to calculate the ef- 
ficiency of the various parts of the joint where possible 
failure may occur. The same line of reasoning, and simi- 
lar methods of operation, may be employed for any kind 
or type of riveted joint that may be used in a steam boiler. 

Size of Rivets and Pitch 

There are no absolute rules for determining the size of 
rivets to be used in any given case, for with different size 
rivets, the same efficiency of joint may be obtained. The 
size to be chosen depends upon several factors and varies 
with any one of them. The things to be considered are: 
the shearing strength of the material of which the rivets 
are made, the tensile strength of the plates to be riveted to- 
gether, the pitch of the rivets, and the type of joint to be 
made. The required efficiency of joint determines the 
type to be used and the greatest pitch of rivets allowable 
depends upon the thickness of plate to be used, the object 
sought being steam-tight joints. 



RIVETED JOINTS 27 

For single-riveted lap joints, the United States Super- 
vising Inspectors of Steam Vessels recommend a rivet 
diameter equal to the plate thickness plus 7/16 in. using 
steel plates and steel rivets. For double-riveted lap 
joints, a rivet diameter equal to the plate thickness plus 
$/& in. Some authorities make the rivet diameter range 
from plate thickness plus 3/8 in. to plate thickness plus 
1/2 in., with plates from 1/4 in. to 1/2 in. thickness. 

For triple-riveted lap joints rivet diameters range from 
plate thickness plus 3/8 in. to plate thickness plus 7/16 
in. with plates from 1/4 to 1/2 in. thickness. 

For double-riveted butt joints, triple-riveted butt joints, 
and for quadruple-riveted butt joints, rivet diameters 
range from plate thickness plus 5/16 in. to plate thickness 
plus 7/16 in. The foregoing is intended to give a general 
idea only of rivet sizes that may be chosen to be some- 
what in proportion to the joint as a whole. In the end it 
is a matter of choosing that size rivet and a certain pitch 
which w r ill give the highest efficiency of joint, consistent 
with steam-tight work, type of joint, strength of materials 
and all other considerations. It is a matter of "cut and 
try" until the best is arrived at. 

Distance Between Adjacent Rows of Rivets 

The distance between adjacent rows of rivets, center to 
center, is sometimes called transverse pitch. When the 
rivets are subjected to the same kind of shear, this dis- 
tance should not be less than twice the diameter of the 
rivets, nor more than two and one-half times the diameter 
of the rivets used. If the distance between the rows of 



28 ARITHMETIC OF THE STEAM BOILER 

rivets is too small, the plate is likely to fracture along a 
diagonal line, or diagonal pitch as it is termed. 

If the transverse pitch is at least equal to twice the 
diameter of rivets, failure of the plate will not occur 
along the diagonal line, but rather in the net section of 
plate along the line of rivets. This makes the calculation 
of joint efficiency somewhat simpler. In cases where the 
outer butt strap is not as wide as the inner strap, the dis- 
tance between the line of rivets in double shear and the 
line in single shear should be two and three-quarters or 
three times the diameter of rivets, in order to have a 
properly formed rivet head, and also room to calk the 
outer strap. 

Safe Working Pressure of Cylinders with Riveted 

Joints 

It w r ill be remembered that formula n, page 9, gives 
the safe working pressure of a cylinder without any 
visible joint. But cylinders having riveted joints must 
be calculated with the efficiency of longitudinal joint 
taken into consideration. The rule will be the same as 
that expressed in formula 11, with the additional factor 
of joint efficiency expressed as a decimal value. 

Let e represent the efficiency of the longitudinal 
seam or joint; the efficiency of the girth seam is not 
required for reasons explained at the beginning of this 
chapter. 

The formula now becomes: 

/X Ty< e 
— r-77 — = safe working pressure per square inch. 



RIVETED JOINTS 29 

Using the same example illustrating formula 11 , page 

11, and assuming a joint efficiency of say 85 per cent. 

the statement becomes: 

.5X50000X .85 

— — — — =141.66 lb. safe pressure per square 

inch. No matter what the efficiency of the joint may be, 
nor by what method it may be found, it is always to be 
applied as shown in the example just given. Of course 
there are other ways of arranging and simplifying the fac- 
tors in the formula, but no matter what the arrangement 
or how simplified, the result will always come out the 
same if the work is correctly done. Abbreviated formulas 
and rules are convenient to those who know of their 
derivation, but they are not satisfying to those who do not 
know just how each simplified factor was obtained. 
For this reason, no attempt is made in this work to ab- 
breviate anything that will detract from the value of 
any problem presented, as far as underlying elements 
and principles are concerned. Any one who understands 
how to calculate the efficiency of riveted joints, and how 
to find the safe working pressure of spherical or cylin- 
drical vessels as given in this work, will be able to work out 
similar problems, no matter in what form they may be 
given, or what rule it may be desired to apply to them. 

So far, the shell and its riveted joints only have been 
considered. The rules given apply to the cylindrical parts 
of all boilers of whatever make. There are other rules 
relating to shells of special design, and these rules will 
be given further on. 

The next in order, at present, is the bracing or staying 
of boiler heads and flat surfaces in boilers. 



CHAPTER II 

Boiler Heads — Unstayed Heads 

Bumped heads may be either convex or concave accord- 
ing as to how placed in a shell. Fig. 6 shows the applica- 
tion of the two forms, (a) being a concave bumped head 
while (b) is a convex head. The arrows show the direc- 
tion of pressure acting against the heads. Bumped heads 
do not require bracing, particularly the convex (b) form 




Fig. 6. — Bumped heads, (a) Concave head, (b) Convex head. 
Arrows show the direction of pressure. 



as it is already in the form that internal pressure would 
tend to make it assume. In the concave (a) form, the 
tendency of internal pressure is to collapse the head, and 
allowance is made for this in the rule which will be given 
presently for the safe working pressure allowed. Bumped 

30 



BOILER HEADS— UNSTAYED HEADS 



31 



heads may be either single or double riveted to the shell. 
It is necessary to know the radius to which a head is 
bumped when making calculations for safe working 
pressure. A bumped head is presumed to be virtually 
part of the surface of a sphere. To find the radius to 
which a head is bumped, take half the diameter of the 
head where it fits into the shell, and multiply that value 
by itself, and divide the product by the height of the 




d= 4 Inches 
Radius =4 Inches 
h =.55 Inches 



do, Horizontal Centex Line 
ef, Vertical Center Line 

d= Diameter of Head, Inches 

/j=Height of Bump, Inches 

O— Center from which the Bump is Struck 
OC ^Radius to which Bump ie Struck 



Fig. 7. — Diagram showing how to find radius of bumped head. 

bump. To the quotient add the height of the bump and 
divide the sum by 2. It is usual to take the dimensions 
in inches. 

To find the radius to which a boiler head is bumped the 
following formula may be used: 

Referring to Fig. (7), 



h 



-= radius. 



32 ARITHMETIC OF THE STEAM BOILER 

All dimensions to be taken in inches. 

To find the safe working pressure of a bumped head like 
that in (b) and when it is single riveted to the shell, the 
following formula may be used: 

* ixs ( \ 

P= ^Xr W 

When the head is double riveted to the shell, then the 
formula becomes: 

P = ^Xr (2) 

When the head is concaved like that in (a) and single 
riveted, then the formula becomes: 







txs 

P = 5Xr 


(3) 




When a 


concaved head is double-riveted the formula 


b 


ecomes: 


txs 


(a1 



* 4iXr 
In these formulas the values of the letters are: 

p —safe working pressure in pounds per square 

inch. 
t = thickness of metal in the head, expressed in 

inches. 
r = radius in inches, to which the head is formed. 
S = tensile strength of the material of which the 

head is made, expressed in pounds per square 

inch section. 



BOILER HEADS— UNSTAYED HEADS 33 

As the foregoing formulas are similar in construction, 
one example will serve to illustrate the operation of all. 

A boiler head is bumped to a radius of 60 in. made of plate .5 in. 
thick, with a tensile strength of 50,000 lb. and double riveted to the 
shell. What working pressure will be allowed? 

The operation is as follows: 

p = — — zn — = 166.66 lb. per square inch. 
2.5X60 

Unstayed Flat Heads 

When a boiler head is flat and not stayed, the following 
formula may be used: 

txs 



P = 



•54X.4 



The letters have the same values as for the preceding 
formulas, and A equals the area of the head expressed in 
square inches. 

Example. — A flat head is 30 in. in diameter, . 75 in. thickness 
of plate having a tensile strength of 55,000 lb. per square inch, 
what pressure will be allowed? 
Operation: 

p = — — — — — — o — = 108 lb. per square inch. 

.54X30X30X.7854 F H 

All the rules for boiler heads of the kind just described 
are those prescribed by the Board of Supervising In- 
spectors of Steam Vessels in the United States. 

If it is desired to find what thickness a bumped head 
should be, having the tensile strength, the radius to 



34 ARITHMETIC OF THE STEAM BOILER 

which the head is bumped, and the pressure in pounds 
per square inch to be carried, it is a matter of transpos- 
ing the terms of the proper formula in the group just 
treated of. 

Suppose in the last illustrative example given that it is 
desired to know what thickness of head should be em- 
ployed. The formula transposed will be: 

S ~ ~ l 

Applying this to the example, the statement becomes: 
166.66X2.5X60 



50000 



-=.5 in. thickness. 



Thickness of Boiler Heads, Massachusetts Rules 

In actual practice, however, the thickness of boiler 
heads is not derived mathematically but empirically. 
The rules in the state of Massachusetts require the thick- 
ness to be as follows: Boilers up to and including 42 
in. diameter, heads must be 3/8 in. From 42 in. to 54 
in. diameter, heads must be 7/16 in. From 54 in. 
to 72 in. diameter, heads must be 1/2 in. Over 72 in. 
diameter, heads must be 9/16 in. 

Thickness of Boiler Heads, Ohio Rules 

The rules formulated for bumped heads by the Board 
of Boiler Rules in the State of Ohio differ slightly from 
those given. 



BOILER HEADS— UNSTAYED HEADS 35 

The minimum thickness of a convex head shall be de- 
termined by this formula: 

RXF.S.XP 
T.S. ~ l 

The minimum thickness of a concave head shall be de- 
termined by this formula: 

RXF.S.XP 

.6(7.5.) 

In these two formulas the values are as follows : 

R = one-half the radius to which the head is 

bumped. 
F.S. =5= factor of safety. 
P = working pressure, in pounds per square 

inch, for which the boiler is designed. 
T.S. = tensile strength, in pounds per square inch, 

stamped on the head by the manufacturer. 
t = thickness of head in inches. 

The radius of head shall not exceed the diameter of 
the shell. 

When a convex or concave head has a manhole open- 
ing, the thickness as found by the formulas just given, 
shall be increased by not less than 1/8 in. 

The minimum thickness of plates in stayed flat surface 
construction shall be 5/16 in. 

The minimum thickness of tube sheets shall be as 
follows: 

" When the diameter of tube sheet is 42 in. or less, the 
thickness is 3/8 in.; over 42 in. to 54 in. inclusive, 7/16 
4 



36 ARITHMETIC OF THE STEAM BOILER 

in.; over 54 in. to 72 in. inclusive, 1/2 in.; over 72 in.. 
9/16 in. 

Stays and Stay Bolts 

The maximum allowable stress per square inch net 
cross-sectional areas of stays and stay bolts as denned 
in the Massachusetts rules, is as follows: 

Weldless, mild steel head to head or through stays, 
8000 lb. for sizes up to and including 1 1/4 in. diameter, 
or equivalent area, and 9000 lb. for sizes over 1 1/4 in. 
diameter or equivalent area. Fig. 8 (a) illustrates 
direct or through stay arrangement. 

Weldless, mild steel diagonal or crow-foot stays, 
7500 lb. for sizes up to and including 1 1/4 in. diameter, 
or equivalent area and 8000 lb. for sizes over 1 1/4 in. 
diameter or equivalent area. 

Weldless, wrought-iron, head to head or through stays, 
7000 lb. for sizes up to and including 1 1/4 in. in diameter 
or equivalent area, and 7500 lb. for sizes over 1 1/4 in. 
diameter or equivalent area. 

Weldless, wrought-iron, diagonal or crow-foot stays, 
6500 lb. for sizes up to and including 1 1/4 in. or equiva- 
lent area, and 7000 lb. for sizes over 1 1/4 in. diameter 
or equivalent area. 

Welded mild steel or wrought-iron stays, 6000 lb. 

Mild steel or wrought-iron stay bolts 6500 lb. for sizes 
up to and including 1 1/4 in. diameter or equivalent area, 
and 7000 lb. for sizes over 1 1/4 in. diameter or equivalent 



area 



When a greater allowable stress per square inch on 



BOILER HEADS— STAYED HEADS 



37 



stays and stay bolts is required than those just given, the 
material shall conform to the following physical qualities: 

The tensile strength shall not exceed 62,000 lb. per 
square inch. 

The yield point shall not be less than one-half the 
tensile strength. 

The elongation per cent, in 8 in. shall not be less 
than 28. 

Direct Stays 

To find the safe working pressure per square inch that 
may be carried by stays of a given size, the following 
formula may be applied: 





Fig. 8. — Arrangement of stays in boiler heads, (a) Direct 
through stays in horizontal rows, seven stays, (b) Diagonal stays 
in concentric rows, seventeen stays. 



aXS 



(1) 



38 ARITHMETIC OF THE STEAM BOILER 

To find the diameter of stays required, 

A -¥=°- - 41k' d (2) 

To find the area supported by one stay, and the dis- 
tance between stays, 

^^=A, and ylA=D. (3) 

P X 

To find the required tensile stress that may be en- 
dured by stays: 

^ = S (4) 

a 

The value of the letters in this group of rules is as 
follows : 

A = area in square inches, supported by one stay. 
a = area, cross-sectional, in square inches, of stays. 
p = pounds pressure per square inch. 
S = tensile strength of stays in pounds per square inch. 
d = diameter of stays in inches. 
D = distance between stays, in inches. 

Example, illustrating the foregoing rules. 

Assume stays of 1 in. diameter at the smallest part, with an 
allowable stress of 6000 lb. per square inch, and distanced 6 in. 
center to center. 

The area of each stay will be i 2 X . 7^54= • 7^54 sq. in. 

The area supported by each stay will be 6X6 = 36 sq. in. 

Applying these values to formula (1) the statement 
becomes: 

.^854X^000 = lb per square inch allowable pressure. 

36 



BOILER HEADS— STAYED HEADS 39 

To formula (2), 
36X130.9. 



6000 



= .7854 sq. in. area of each stay. 



and, \~~~i — =I * n * diameter of stays. 

To formula (3), 

.7854X6000 . 111 

- J — = 36 sq. in. area supported by each stay. 

and, V 36 = 6 in. distance center to center of stays. 

To formula (4), 

z — —=6000 lb. tensile stress allowed on stays. 

•7854 

In the foregoing no allowance has been made for the 
space occupied by the stays in the sheets supported. 
This is on the side of safety and is generally accepted. 
If in any case it is not accepted, it becomes a matter 
of subtracting the area occupied by the stays from the 
area as found from the center to center measurement. 

Diagonal Stays 

The size of a diagonal stay depends upon the angle it 
makes with the surface it is helping to support, when con- 
sidered in relation to a direct stay. The less the angle is, 
the larger in diameter must the stay be. The nearer 
a stay is to being at right angles to the surface it supports, 
the smaller in diameter it may be. The same principle 



40 



ARITHMETIC OF THE STEAM BOILER 



applies to any form of stay that may be used, other than 
a circular cross-section. 

In Fig. 9 is shown an ordinary diagonal stay attached 
to the boiler head at C and to the shell at E. The length 
of the stay is considered as CE; the distance CD also < 
enters the calculation as will be shown presently. 





Fig. 9. — Forms of attachment of diagonal stays, (a) Riveted 
at both ends, (b) Riveted at one end, nuts and washers at the 
other end. 

To find the area of a required diagonal stay, first find 
the area of a direct through stay, as has been explained. 
Call the length CE of the required diagonal stay, x and 
the distance, CD, y. 



BOILER HEADS— STAYED HEADS 41 

Let a = the area of direct stay. 
Let A = the area of diagonal stay. 
Let S = tensile strength of stay. 



Then, 



aXx . , N 
= A (1) 

y 

AXy 



a 

AXy 

x 

aXx 



= x (2) 

=a (3) 

=y (4) 



x A ° Safe pressure per 

area supported by = sc l ua, ; e inch . ^ 
one stay ma ^ be carned - 



(5) 



Example, illustrating the foregoing group of rules. 

Assume that the area of a direct stay has been found to be .7854 
sq. in. (due to 1 in. diameter as before taken). That the length 
of the required diagonal stay is to be 36 in. and that the distance 
between perpendiculars of points of attachment is 30 in. Then, 
applying these values to formula (1), 

1 = .9425 sq. in., nearly, cross-sectional area of diagonal 

stay required, and the corresponding diameter is: 



V 



.9425 . .. 

— - — = 1 .095 in. diameter. 

•7^54 



The nearest commercial size brace that would be used is 1 1/8 
in. diameter. 



42 ARITHMETIC OF THE STEAM BOILER 

To formula (2), 

jt = 36 in., length of the required diagonal stay, 

as measured on the line CE in Fig. 9. 
To formula (3), 

7 = .7854 sq. in., area of direct or through stay. 

To formula (4), 
.7854X36 



= 30 in., distance between perpendiculars as 

measured on the line CD in Fig. 9. 

To formula (5), 
.9425X30 



X6000 



= 96.1 lb. per square inch pressure 

allowed. 

Other formulas for obtaining the diameters of direct 
and diagonal stays. 

For direct stays, 



/SX.7854 
For diagonal stays, 



, / AXp , v 

d Hsx^I (I) 



/ xXAXp () 

\yXSX. 7854 K 

The values of the letters are the same as those used for 
the stay group of formulas. 

Examples, illustrating these formulas. 

Assume 36 sq. in. area supported by one stay (pitch, 6 in. center 



BOILER HEADS— STAYED HEADS 43 

to center), pressure to be carried, 100 lb. per square inch, 6000 lb. 
allowed stress per square inch on stays, then to find the diameter of 
direct stay as stated in formula (1) the statement becomes: 



/ 36XIOO , V 

<f = \6oooX.78S4 = - 873S1I1 - ; (I) 

in practice 7/8-in. diameter stays would be used. 

For formula (2) diagonal stays, assume same values, and in addi- 
tion, that the length of the diagonal stays are 36 in., and that the 
horizontal distance between the ends of stays — as shown in Fig. 
9 — is 30 in., the statement becomes: 



■Ato 



36X36XIOO . ( v 

in practice i-in. diameter stays would be used in this case. 

The area in square inches supported by one stay 
multiplied by the number of pounds pressure per square 
inch carried in any given case gives the load in pounds 
the stay must sustain. 

The cross-sectional area in square inches of each stay 
in any group, multiplied by the allowable tensile stress 
per square inch section, gives the allowable load in pounds 
the stay may sustain with safety. 

The load, in pounds, on a stay divided by the cross- 
sectional area of the stay in square inches will give the 
stress in pounds per square inch. 

To find the number of stays of a given size required 
to support a given area, multiply the area of the sheet 
to be stayed by the steam pressure in pounds per square 
inch to be carried, and divide the product by the total 
allowable stress for the given size of stay. 



44 ARITHMETIC OF THE STEAM BOILER 

Example. — Assume i i/8-in. round iron stay bolts with an allow- 
able stress of 6000 lb. per square inch, and an area of 2000 sq. in. 
to be supported against a pressure of 90 lb. per square inch. How 
many stays are required? 

(1 i/8) 2 X. 7854X6000 =5964 lb. 

allowable on each stay. Then: 

2000X90 . 
=30. 18, say 31 stays will be required. 

Rivets Securing Stays 

The combined cross-sectional area of the rivets which 
secure the stays to boiler heads must not be less than that 
of the stay itself. The rivets securing a diagonal stay 
to the head are in tension chiefly, but they also endure a 
certain amount of bending. Those rivets which secure 
the end of the stay to the shell are in single shear chiefly, 
and to a certain extent are in tension. In practical work 
the rivets used for both ends of such stays are of the same 
size, a high factor of safety being used to make due allow- 
ance for the different stresses, and the difference in 
value of rivets in tension and in shear. It is considered 
advisable to allow but 4000 lb. per square inch section 
for rivets used on diagonal stays, in order to be on the 
safe side. 

The part of a head above the tubes to be braced is 
a segment of a circle. In laying out the position of the 
stays it will be found that they cannot be arranged so 
that exactly the same load will be borne by all. It is 
customary to arrange the stays in concentric rows, as 
shown in Fig. 8 (b). 



BOILER HEADS— STAYED HEADS 45 

It is usual to consider that the flange of a boiler head 
supports the head for a distance of at least 3 in., measured 
from the inside of the flange. With modern methods of 
making plates and flanging them, the radius to which a 
head is flanged is now greater in proportion to the thick- 
ness than used to obtain, and it is thought by some that 
more than 3 in. may be counted upon as being supported 
by the flange, depending on the thickness of the head, 
the pressure to be carried, and the disposition of the stays. 
Speaking in general, the 3-in. distance is reasonable and 
safe to use. 

The tubes that are expanded into the tube sheet sup- 
port that part of the head. A certain portion of the 
head above the top row of tubes is supported by them, 
depending on the size and holding power of the tubes 
themselves. 

A reasonable and safe distance to consider in such calcu- 
lations is that of one-half of the bridge between the tubes. 
Suppose, for example, that in a given boiler the tubes are 
31/2 in., spaced 4 3/4 in. center to center; the bridge, or 
section of plate between the tubes is 1 1/4 in. ; then one- 
half that distance or 5/8 in. may be considered as being 
supported above the top row of tubes by the tubes in the 
top row. 

Diagonal braces of the crowfoot type usually have two 
rivets spaced 4 in. center to center. This permits of a 
proper spacing of stays, which is the main point to con- 
sider in laying out a head. The maximum allowable 
pitch must not be exceeded. The braces can be made to 
suit the load. Formula (7) decides just what pitch may 
be used. 



46 ARITHMETIC OF THE STEAM BOILER 

Flat Surfaces to be Stayed, in which the Thickness 
of Plate Enters the Calculation 

112 X^ 2 

A = t — > for plates up to 7/16 in. thick. (1) 

V 

120X/ 2 

A= — ? for plates above 7/16 in. thick. (2) 

P 

140 X^ 2 
A= — - — ? for screw stay bolts and nuts. (3) 

112 X^ 2 
p = — -j — 7 for plates 7/16 in. and under. (4) 

120X/ 2 
p= — -j , for plates over 7/16 in. thick. (5) 

T AO X / 2 

p = — -j — -> for screw stay bolts and nuts. (6) 



/112 X£ 2 
S = \ > f° r plates 7/16 in. thick and less. (7) 

/l20X^ 2 

S = \l f ioi plates above 7/16 in. thick. (8) 

S = \l— j for screw stay bolts and nuts. (9) 

-J*M (I0 ) 

\ 112 



t 



If this formula gives more than 7/16 in. for the value of 
/, use the next formula with the factor 120 in it. 



=J^4 (II) 

\ I20 

\ 12" 



[40 



BOILER HEADS— STAYED HEADS 47 

In the foregoing group of formulas, the values are as 
follows : 

A = area in square inches, supported by one stay. 

The pitch can be found by extracting the square 

root of A. 
t — thickness of plate, expressed in sixteenths of an 

inch. Example: if the plate were 7/16 in. 

thick, / in this case would be 7. In other 

words, t is the numerator of the fraction 

whose denominator is 16. 
p = pressure in pounds per square inch, allowed 

to be carried. # 

5 = pitch of stays, center to center. 

112, 120, 140 are constants used in the respective 
formulas. 

Examples, illustrating the application of the formulas in this 
group: 

Assume plate 7/16 in. thick; pressure to be carried, 100 lb. per 
square inch. Required, area that may be supported by one stay, 
from which the pitch distance may be found. 

Then, in formula (1), 

112 X 7 2 
A = = 54.88 square inches supported by one stay. 

In formula (4), 

112 ^^ 1 
p = «x— =100 lb. per square inch pressure allowed. 

In formula (7), 

/112 X 7 2 

S = % — = 7.4081 in. center to center of stays. 

\ 100 



4 8 



ARITHMETIC OF THE STEAM BOILER 



In formula (10), 

t=\ — — = 7, numerator of the fraction 7/16, 

\ 112 

thickness of plate in inches. 

Formulas (4), (5), and (6) in this group serve as a check 
on the design of stays, for by applying the proper one 
to any given case it can be determined as to whether or 
not the desired pressure may be carried. 

Segments to be Braced 

In Fig. 10 is shown that portion (shaded) of a boiler 
head that requires to be braced. The flange is presumed 




i/=Height of Segment in Inches 

R= Radius of Circle of which the Segment is a Part 

C = Center of the Circle of which the Segment is a Part 

Fig. 10. — Segment of head to be braced. 



to support the head for a distance of 3 in., and the tubes, 
a distance of 2 in. above as shown in the figure. These 
are the generally accepted figures by most designers of 
boilers and experience seems to have proved them to be 
safe. 



BOILER HEADS— STAYED HEADS 49 

To find the height of the segment requiring bracing, 
subtract 5 in. from the distance from the tubes to the 
highest part of the shell. To find the diameter of the 
circle of which the segment is a part, subtract 6 in. from 
the diameter of the boiler. 

To find the area of a segment sufficiently close for all 
practical purposes, the following formula may be used 
(see Fig. 10): 



4 



4XH* 2XR , 

X a /— rr .608 = area. 



In which H = the height of segment, R = the radius of 
circle, and the numerals, constants. 

Suppose, for example, it is desired to know the area 
of a segment of a circle whose diameter is 50 in., the 
height of the segment being 20 in. The statement 
becomes: 



4x^o! x i,_x*s_ 6o8= 

3 \ 20 



1600 co 

3 \20 

1600 



XaJ 2 '5-- 6 ° 8 : 



*22 x Ji.892_ 
3 x 

1600 



3 
1600X1.37S 



<i-375 

= 733-3 sq. in. area. 



50 ARITHMETIC OF THE STEAM BOILER 

The formula just given, when applied to segments 
whose heights are three-tenths of the diameter of the 
circle of which the segment is a part, gives results correct 
within i/io of i per cent. 



Method of Finding the Area oe a Segment by Use 
oe a Table 

As well as the formula that has been given and 
illustrated by an example, the area of a segment may be 
found by using the table on page 51, and by an approxi- 
mation method illustrated in Fig. 11. 

It is desired to find the area of the shaded segment of 
the circle in Fig. 11, by using the table. The height 
of the segment is 16 in., and the diameter of the circle of 
which the segment is a part is 54 in. The method of 

operation is as follows: — = .2963, which is the height 

54 
of a similar segment of a circle whose diameter is 1.0. 

The two nearest values, Ht. column in the table, to .2963 

are .29 and .3. The corresponding areas are, for .29 = 

.18905; for .3 = .i98i7. 

A convenient value to use without going into inter- 
polation is .2963 = . 195, and S4 2 X. 195 = 568.620 sq. in. 
area. 

As a check on the foregoing, the formula may be 
applied to the same example: 



-X-v/—— .608= 567.780 sq. in. area. 
3 \i6 



BOILER HEADS—STAYED HEADS 



51 







AREAS OF 


CIRCULAR SEGMENTS 






Ht. 


Area 


Ht. 


Area 


Ht. | 


Area 


Ht. 


Area 


Ht. 


Area 


O.OI 


0.00133 


O.II 


0.04701 


0.21 


0.1199 


0.31 


0.20738 


0.41 


0.30319 


0.02 


0.0037s 


0.12 


0.5338 


0.22 


0.12811 


0.32 


0.21667 


0.42 


0.31304 


0.03 


0.00687 


0.13 


0.06 


0.23 


0.13646 


0.33 


0.22603 


0.43 


0.32293 


0.04 


0.01054 


0.14 


0.6683 


0.24 


0.14494 


0.34 


0.23547 


0.44 


0.33284 


0.05 


0.01468 


0.15 


0.7387 


0.25 


0.15355 


0.35 


0.24498' 


o.45 


0.34278 


0.06 


0.01924 


0.16 


0.08111 


0.26 


0.16226 


0.36 


25455 


0.46 


0.35274 


0.07 


0.02417 


0.17 


0.08854 


0.27 


0.17109 


o.37| 


0.26418 


0.47 


0.36272 


0.08 


0.02943 


0.18 


0.09613 


0.28 


0.18002 


0.38 


0.27386 


0.48 


0.3727 


0.09 


0.03501 


0.19 


0.10390 


0.29 


0.18905 


0.39 


28359 


0.49 


0.3827 


O.I 


0.04087 


0.2 


0.11182 


0.3 


0.19817 


0.4 


0.29337 


0.5 


0.3927 




Fig. 11. — Approximate method of rinding the area of segment 
of a circle. Area of A B C D, 11X53 = 583 square inches. Area 

of semicircle, : = 1145.115 square inches. Then, 1145.115 

-583 = 562.115 square inches, area of shaded segment. 



By the table, 
By the formula, 
Difference, 

Difference per cent. = 



568.620 

. 840 sq. in. 
.840 



567.780 



Xioo = .i48 



52 ARITHMETIC OF THE STEAM BOILER 

which is sufficiently close for all practical purposes in 
boiler work. 

The table gives the areas of segments from' a height of 
.01 up to .5 (a semicircle) increasing by hundredths. 
The areas of segments are from a circle of unit diameter. 
If the values in the table are taken as feet, they apply to 
a circle 1 ft. in diameter; if the values are taken in inches, 
they apply to a circle 1 in. in diameter. 

An approximate method is illustrated also in Fig. 11. 
Consider the length of the rectangle ABDC as 1 in. less 
than the diameter of the circle, and the width as n in. 
This gives an area of 53X11 = 583 sq. in. This is to be 
subtracted from the area of the semicircle of which the 
segment is a part. 

The area of a 54-in. circle is 54 2 X. 7854= 2290.23 sq. 

r 1 • • 1 2200.23 
in., and of the semicircle, =1145.115 sq. in. 

From this subtract 583, which gives 562.115 sq. in. 
as the area of the segment by this method. Comparing 
this answer with that obtained by the formula: 

567.78 — 562.115 = 5.665 sq. in. difference. 

The difference per cent, is: 

5-665 
56^ Xl00 = -"' 

or 1 per cent, in round numbers, for this particular case. 
It must not be thought that the approximate method 
just described will always differ from that of the formula 
by 1 per cent. The difference may vary as much as 2 
per cent., according as to the size of the circle and the 
height of the segment as ordinarily found in stationary 



BOILER HEADS— GIRDER BARS 



53 



boiler practice. In circles having a diameter from 30 
in. to 102 in., the result found by the approximate method 
will be in error less than 2 per cent., when the height of 
the segment is not less than three-tenths the diameter 
of the circle. The Hartford Steam Boiler Inspection 
and Insurance Company sanction the use of the approxi- 
mate method, because of its simplicity, and because it 
gives results sufficiently close for all practical purposes 
in relation to segments of horizontal tubular boiler heads 
to be braced. 

Girder Bars 

Girder bars are of two kinds, the split bar, Fig. 12 (a), 
and the solid bar (b). The figure shows the application 



3 Bolts 

ftl ftl tfl 




Split, or Double Bar 

(a) 





Solid Girder Bar 

(b) 



Fig. 12. — Girder bars. 



of girder bars to the combustion chamber tops of marine 
boilers and to the crown sheets of locomotive boilers. 

The safe working pressure for solid girders, Fig. 12 (6), 
may be found from the following formula: 

CXd 2 Xt 



P = 



(w-p)XDX? 



54 ARITHMETIC OF THE STEAM BOILER 

in which the value of the letters are: 

P= safe working pressure in pounds per square 
inch. 

d= depth of the girder in inches. 
/= the thickness of girder in inches. 

w= width in inches of combustion chamber, meas- 
ured in direction at right angles to the length 
of the bar. 

p= pitch of bolts in inches, securing bars to the 
sheet. 

D= distance in inches, from center to center of 
girders. 
1= length in feet, of girders. 

C= a constant, according to design, as follows: 
550 for bars with one bolt, 825 for bars with 
two or three bolts, 935 for bars with four 
bolts. 

The safe working pressure when split girder bars, Fig. 
12 (a), are used, may be found from this formula: 

24oooX/X ^ 2 

In which the value of the letters are: 

P= safe pressure in pounds per square inch. 
/= thickness in inches of one girder. 
d= depth in inches of one girder. 
D= distance in inches, from center to center of 
girders. 
1= length in inches of girders. 
24,000= a constant. 



BOILER HEADS— GIRDER BARS 55 

The following example shows the application of the 
first formula. 

A solid rectangular steel girder bar 3 ft. long, 8 in. deep, 
2 in. thick, three bolts, bars spaced 8 in. center to center, 
pitch of bolts 8 in., and combustion chamber top 32 
in. wide. What pressure may be carried? 

In this example the value of C is 825. Then: 

825X8 2 X2 105600 D „ 
r=-, ^-- , o w = ~z~= 183.3 lb. per square inch 

The values quoted for C are those prescribed by the 
United States Board of Supervising Inspectors of Steam 
Vessels. 

The following example shows the application of the 
second formula. 

A split or double girder bar, 8 in. deep, 1 in. thick, 30 
in. long, the bars being placed 8 in. center to center. 
What pressure may be carried? 

24000X1X8 2 1536000 



8X30 2 7200 



= 213.3 lb. per square inch. 



The strength of stay bolts, and the load carried by each 
one fitted to girder bars, are calculated in the same way as 
for any flat surface; and the same rules apply for the 
maximum pitch and safe working pressure as for any 
flat surface. 



CHAPTER III 

Manhole Reenforcing Rings 

Reenforcing rings are for the purpose of strengthening 
manhole openings in boilers. When only one ring is 
used it may be placed either inside or outside of the shell. 
When two rings are used, one is placed inside and the 
other outside the shell. 

• Single rings may be from i 1/4 to 1 1/2 times the thick- 
ness of the shell plates, and when double rings are used 
the thickness of each may be the same as that of the 
shell plates. The rings must be riveted to the shell by 
a sufficient number of rivets of proper size, so that the 
combined resistance to shearing will be at least equal to 
the resistance of the rings to tensile stress. The various 
formulas for reenforcing rings are as follows: 

W = — — +D. (1) For single-riveted rings. 

2 X t 

W= — —+2X0. (2) For double-riveted rings. 

2 X t 

Formulas (1) and (2) are for single rings. 

Z/X/ 
W= -. +D. (3) For single-riveted rings. 

4Xf 

Z,X / 
W = —rz I + 2XD. (4) For double-riveted rings. 

4X£ 

56 



MANHOLE REENFORCING RINGS 57 

Formulas (3) and (4) are for double rings. 
The values of the letters are as follows: 
W= width of rings in inches. 
t= thickness of rings in inches. 
t\ = thickness of shell plates in inches. 
D— diameter of driven rivet in inches. 
L = length of opening in shell in inches. 
In calculating the least number of rivets to be em- 
ployed, the net section of the ring is to be used. In 
single riveting, the net section is found by subtracting 
the diameter of the rivet hole from the width of the 
ring, and then by multiplying the remainder by the 
thickness of the ring. When the ring is double riveted, 
subtract twice the diameter of the rivet holes from the 
width of the ring, and multiply the remainder by the 
thickness. The diameter of rivet holes is the equivalent 
of the driven size of rivets, and refers to the trial size of 
rivets to be chosen in the following formulas: 

at 4XTXA . 

iV = 5X(^X.78 S4 ) (5) F ° r Smgh rmgS * 

N = i.8 S X5X(^X. 7 8S4) (6) DaMe ringS - 

The values of the letters are: 

N= number of rivets required. 
A = net section of ring in square inches. 
T= tensile strength of ring per square inch section. 
S = shearing strength of rivet per square inch section. 
D = trial diameter of driven rivet in inches. 
.7854 = a constant used in finding circular areas. 



58 ARITHMETIC OF THE STEAM BOILER 

4 = a constant used for single reenforcing rings. 
8 = a constant used for double reenforcing rings. 

If either of these formulas give a too small number of 
rivets, too widely spaced, a smaller trial diameter of 
rivet must be chosen and the formula again applied. If 
on the other hand too many rivets are found, causing 
them to be too close together, a larger trial diameter of 
rivet must be chosen. 

Examples showing the application of the foregoing 
group of formulas. 

A manhole in a boiler is 11X15 in.; the n -in. dimen- 
sion lying in the direction of the length of the boiler. 
The shell plates are 3/8 in. thickness, and the reenforc- 
ing ring is to be 1/2 in. thick, single riveted, driven size 
of rivets 7/8 in. What is the required w T idth of ring? 

Using formula (1) the statement becomes: 

I T ^ 9*7 f 

W = ' " ' ^ +-875 = 5.0 in., answer. 

2 X . 5 

A manhole opening is 11X15 in.; the n-in. dimension 
lying in the direction of the length of the boiler. The 
shell plates are 1/2 in. and two 1/2-in. rings are to be 
used, double riveted, driven size of rivets 7/8 in. What 
width of rings is required? 

In this case formula (4) is to be used. The statement 
becomes: 

1 1 X ^ 
W = +2X .875=4.50 in., answer. 

4X .5 

How many 7/8-in. driven rivets are to be used in a 
single ring 1/2 in. thick, 5 in. wide, 60,000 lb. tensile 



MANHOLE REENFORCING RINGS 59 

strength, 38,000 lb. shearing stress of the rivets, ring to 
be single riveted. 

Formula (5) is to be used, and statement becomes: 
The net section of the ring is (5 — .875) X. 5 = 2.0625 sq. in. 

4X60000X2.0625 . 

N = -5 77T~6 — 2^7 — o — ^ = 21 required, say 20. 

38oooX(.875 2 X.7854) H J 

When the total number of rivets is an odd number, 
change it so that there will be an equal number of rivets 
on each side of the center. A number divided by 4 may 
be used. 

How many i-in. driven rivets should be used in a 
manhole reenforced by two rings 3/4 in. thick and 4 1/2 
in. wide, single riveted, 38,000 lb. shearing stress per 
square inch section, and 60,000 lb. tensile strength of 
the material of which the rings are made. 

Formula (6) is to be used. The net section of the ring 
is (4. 5-i)X. 75 = 2-625 sq. in. 

_ r 8X60000X2.625 

l\ = — Q . . v . , 9Ky — o — r = 22.8, say 24 rivets. 

i.85X38oooX(i 2 X.7854) ' J ^ 

(1 . 85 is a constant for rivets in double shear.) 

Rings whose width is found by the formulas in this 
group will have a total net cross-sectional area equal to 
that of the metal taken from the shell to make the 
opening. The total net section means for a single ring, 
twice the net sectional area of the ring, and for double 
rings, four times the net sectional area of one ring. 

The formulas for finding the number of rivets to be 
used make the resistance of the rivets to shearing equal 
to the resistance to tensile stress of the metal in the rings. 



6o 



ARITHMETIC OF THE STEAM BOILER 



The constant 4 that appears in formula (5) is obtained 
from two sectional areas for each half of the ring, and 
all the rivets in each half. Therefore in both halves 
of the ring, and when considering the total number 
of rivets in the whole ring, there are four sections to 
consider. 




Fig. 13. — Reenforcing ring. Diagram illustrating direction of 
stress in reenforcing rings, and the origin of the constants 4 and 8 
which appear in the formulas. Arrows a and b show direction of 
stress borne by each half of the ring, and the rivets in each half. 

With two rings, there are four sectional areas for each 
half of the two rings, or twice as much as with one ring, 
therefore the constant becomes 8 in formula (6). 



Heating Surface or Boilers 

The heating surface of a steam boiler is that part 
exposed to the action of the hot gases from the furnace. 
Finding the heating surface is a matter of mensuration, 
and it is expressed in square feet. 

To find the heating surface of a return-tubular boiler, 
multiply two-thirds of the circumference of the shell by 



HEATING SURFACE 61 

the length, both in inches; multiply the number of tubes 
by the circumference of one tube and by its length, also 
in inches; take two-thirds of the area of each tube sheet 
minus the area due to the tube openings in square inches, 
add the products together and divide the sum by 144 to 
convert into square feet. 

To find the heating surface of a vertical tubular boiler, 
multiply the circumference of the fire box by its height 
above the grate, both in inches; multiply the number of 
tubes by the circumference of one and by its length, also 
in inches; find the area of the lower tube sheet minus the 
area of tube openings, all in square inches; add the re- 
sults together, and divide the sum by 144 to convert into 
square feet. 

To find the heating surface of water-tube boilers, 
multiply the number of tubes by the circumference of 
one and by its length, both in inches; find the exposed area 
in square inches of one set of headers; find the number of 
square inches in one-half of the steam drum or drums as 
the case may be; add the values together, and divide by 
144 to convert into square feet. 

The true heating surface of a tube is the side exposed 
to the hot gases; the inner surface in a fire tube, and the 
outer surface in a water tube. 

The following example illustrates the rule for finding 
the heating surface of a steam boiler. 

Example. — What is the total heating surface of a horizontal 
return tubular boiler 60 in. diameter and 1 2 ft. long, with 80 tubes 
2 in. diameter? 

For the sake of simplicity consider the2-in. dimension as the 
inside diameter of the tubes. 



62 ARITHMETIC OF THE STEAM BOILER 

The statement will be: 
Circumference of the shell = 60X3.1416 = 188.496 in. 

Length of shell 12X12 = 144 in. 

Heating surface of the shell 188.496 X144 X 2/3 = 18095.616 

sq. in. 

Circumference of one tube 2X3.1416 = 6.2832 in. 

Heating surface of all the tubes 80X144X 

6.2832 = 72382.46 sq. in. 

Area, in square inches, of one head 6o 2 X. 7854 = 2827.44. 

Two-thirds area of both heads 2/3X2X2827.44 = 

3769.92. 
Area, in square inches, through all the tubes 2 2 X. 7854X80 

Total heating surface ^5-6i6 + 72345-6 + 3769^-2 X 251.328 

144 

9 37o8-48 
= —TT A — =650. 75 sq.ft. 
144 

Grate Area 

The required grate surface in any given boiler depends 
upon the rate of combustion of the fuel, the quantity of 
water evaporated per pound of the fuel used, and the 
total weight of steam generated per hour. 

To find the grate area required for any given plant: 

W 

A = w in which the values are: 
rXw 

A = area of grate surface in square feet. 

W = weight of steam required per hour. 

w = pounds of water evaporated per pound of 

fuel consumed. 
r = rate of combustion in pounds per square foot 

of grate per hour. 

Example. — A battery of boilers is required to generate 7000 lb. 
of steam per hour, consuming 15 lb. of coal on each square foot of 



FURNACES 63 

grate per hour. Assume the evaporation to be 7 lb. of water per 
pound of the coal used, what grate area will be required? 
Applying the formula, the statement becomes: 

7000 
A =^— — = 66.66 sq. ft. 
1SX7 

As the average evaporation per pound of fuel is differ- 
ent for different boilers, the formula given is approximate 
only, but for practical work it is considered suitable to 
use. 

Corrugated Furnaces 

To find the safe working pressure of steel corrugated 
flues and furnaces: 

p = CXT 
D 

In which: 

P= safe pressure in pounds per square inch. 

T= thickness of metal of which the furnace is 

made. 
D= mean diameter of furnace in inches. 
C= a constant, as follows: 
15,600 for Morrison flues, under United States 

Rules. 

14,000 for Morrison flues under British and Canadian 

Rules, and also for Purvis, Fox, and Brown 

flues, under United States, British, and Canadian 

Rules. 

The mean diameter of a Morrison flue, under United 

States Rules, is the least inside diameter of flue plus 



64 ARITHMETIC OF THE STEAM BOILER 

2 in. Under British and Canadian Rules, the mean 
diameter is the diameter at the bottom of the corruga- 
tions, as measured from the outside. The Fox type is 
measured in the same way for the mean diameter by 
the United States, British, and Canadian Rules. 

Example. — What is the safe working pressure on a Morrison 
furnace flue 36 in. mean diameter, with a thickness of metal of 3/8 
in. The value of C in the question is to be taken as 15,600. 

Applying the formula: 

1 5600X .375 , 1K 
P = — = 162.5 lb. 

For furnaces other than corrugated, different authori- 
ties give different formulas, according to size and design. 
For plain flues made in sections not more than 8 ft. in 
length, and with the ends of each section flanged and 
riveted, with a ring between the flanges, this rule may be 
used: 

89600 XT 2 
LXD 
In which : 

P= safe working pressure in pounds per square 

inch. 
T = thickness of flue in inches. 
D= diameter of flue in inches (outside diameter). 
L = length of section in feet. 

Example. — A plain furnace flue is 24 in. diameter and 3/8 in. 
thick, and made in sections of 6 ft. long. What pressure is safe to 
carry? 

Applying the formula: 

89600 X. 3 75 2 „ 1K 

P= TTZ = 105.0 lb. 

6X20 ° 



TUBES 65 

The collapsing pressure of steel tubes of sizes from 3 
to 10 in. may be found from the following formulas. 



P = 86,670-^-- 1386 (1) 



And P=iooo( 1 — yl 1 — 1600^ ) 



(2) 



The first one is to be used when the value of P is greater 
than 581 lb. The second one is to be used when the value 
of P is less than 581 lb. 

The values of the letters in both formulas are: 

P = collapsing pressure in pounds per square inch. 

T = thickness of tube in inches. 

D= outside diameter of tube in inches. 

These formulas are based on experiments conducted 
at the National Tube Works, McKeesport, Pa. 

The factors of safety that may be used for tubes varies 
from 4 to 7, according to surrounding conditions. For 
instance, in a case where considerable damage to life and 
property might result from the collapsing of a tube, a 
factor of safety of 7 should be used. 

Example. — What is the presumed collapsing pressure of a 3. 5-in. 
lap-welded steel tube of .12-in. thickness? What pressure may be 
safely carried where a moderate amount of loss would occur from 
the failure of a tube? 

First try formula (1). 

P=86,67oX~- 2 -i386 
= (86, 670X. 0343) — 1386 = 1586.781 lb. per square inch. 



66 ARITHMETIC OF THE STEAM BOILER 

Formula (i) proves to be the correct one to use, as the result is 
greater than 581 lb. specified. 

The safe pressure to carry, using 5 as a factor of safety under the 
conditions stated, will be: 

1586.781 

= 3i7+ lb. per square inch. 

Under United States Rules, lap-welded boiler tubes 
from 1 in. to 6 in. ; inclusive, may be of any length, and 
may be allowed an external safe working pressure up to 
and including 225 lb. per square inch. This gives a 
liberal factor of safety. 

Horse-power of Boilers 

In reality there is no such thing as horse-power of 
steam boilers, but the term has come into use and there- 
fore requires definition. In order to have a definite value 
by which to compare boiler performances under different 
conditions, the American Society of Mechanical Engineers 
decided that a standard boiler horse-power should be 
equal to the absorption of 33,330 B.t.u. by the water in 
the boiler. This is based on the evaporation of 34 1/2 
lb. of water per hour from a temperature of 212 F. into 
steam of the same temperature and corresponding pres- 
sure, that of the atmosphere. 

As the latent heat of steam at atmospheric pressure is 
required to convert 1 lb. of water at 212 F. into steam of 
212 F., then the latent heat in B.t.u. 's times the number 
of pounds of water chosen as the standard, gives the 
value, thus: 

966.1X34.5=33^30 B.tu. 






BOILER HORSEPOWER 67 

The standard boiler horse-power is found by the fol- 
lowing formula: 

nr _ WX(H-t+ 3 2) 
3333o 
In which, H.P. = the horse-power. 

W = weight of water in pounds actually 

evaporated per hour. 
H= total heat of steam above 32 , at 
the pressure of evaporation. 
/= the temperature of the feed water. 

Example. — A boiler evaporates 3000 lb. of water per hour from 
a feed-water temperature of ioo° F. into steam of 85 lb. gage pres- 
sure, what is the standard horse-power? 

The absolute steam pressure is 85 + 15 = 100 lb. in tound numbers. 
Referring to a table of properties of saturated steam, it will be found 
that the total heat of steam at 100 lb. absolute pressure is, in round 
numbers, 1182 B.t.u. The statement becomes: 

tj D 30ooX(n82-ioo+32) 

H.P. = = 1 00 -f- horse-power. 

33330 

Even values have been taken instead of exact ones, in 
order to simplify the operation. It must also be remem- 
bered that different steam tables in use will give slightly 
different values, but for purely practical work, the dif- 
ference need not be considered. 

Horse-power rating of boilers is sometimes based upon 
the number of square feet of heating surface, varying from 
6 to 20 square feet per horse-power, according as to the 
type of boiler as follows: 

Water-tube boilers, 10 to 12 sq. ft. per horse-power. 
Return tubular boilers, 12 to 15 sq. ft. per horse- 
power. 
6 



68 ARITHMETIC OF THE STEAM BOILER 

Vertical tubular boilers, 15 to 20 sq. ft. per horse- 
power. 

Flue boilers, 8 to 12 sq. ft. per horse-power. 

Plain cylinders, 6 to 10 sq. ft. per horse-power. 
These values are arbitrary and serve only as a guide in 
buying and selling. Different manufacturers had in the 
past adopted different values, there being no fixed stand- 
ard until recently, when the Boiler Manufacturers' As- 
sociation adopted 10 sq. ft. of heating surface to a 
horse-power, in horizontal tubular boilers. 

Ratio of Heating Surface to Grate Area 

The ratio between the heating surface and grate sur- 
face varies with the type of boiler and also with the rate 
of combustion. 

^ . heating surface 

Ratio = — - — 7 

grate surface 

Water-tube boilers, 35 to 40. 
Horizontal tubular, 25 to 35. 
Vertical boilers, 25 to 30. 
Locomotive boilers, 50 to 100. 
Flue boilers, 25 to 35. 
Plain cylinder boilers, 12 to 15. 

Equivalent Evaporation 

Equivalent evaporation from and at 2i2°F. means 
the quantity of water that would be converted into steam 
of 212 F. and at atmospheric pressure, from a feed-water 
temperature of 212 F. as compared with the actual 
evaporation under certain conditions, for any given case. 



Ratio = 



EQUIVALENT EVAPORATION 69 

Equivalent evaporation reduces actual evaporation to a 
standard basis, from which comparisons can be made in 
boiler trials. 

The formula is as follows: 

w _ wX(H-t+ 3 2) 
970.4 
In w T hich the letters have the following values: 

W = equivalent evaporation from and at 212 F. in 

pounds. 
w = actual evaporation in pounds. 
H = the total heat of steam above 3 2 F. at the pres- 
sure of evaporation, as found in the steam tables. 
t = the temperature at which the feed water enters 
the boiler; 970.4 latent heat of steam at atmos- 
pheric pressure according to Marks and Davis's 
tables. 

Example, illustrating the application of the formula. 

A certain boiler generates 3000 lb. of dry steam per hour at a 
pressure of 150 lb. gage from a feed- water temperature of 200 F. 
What quantity of water would have been evaporated had the feed 
water been delivered at 212 F. and converted into steam of 212 F. 
and at atmospheric pressure? 

According to Marks and Davis's steam tables, the 
total heat of steam at 150 lb. gage pressure, or 165 lb. 
(in round numbers) absolute pressure is 1195.0 B.t.u. 
Applying the factors, the statement becomes: 

W = =3175 lb., closely. 

If steam tables be used other than Marks and Davis's, 
slightly different results will be obtained. Before these 
tables were devised, the constant used was 965.7 and 



70 ARITHMETIC OF THE STEAM BOILER 

sometimes 966.1 instead of 970.4. These values repre- 
sent the latent heat of steam at atmospheric pressure, 
according to the tables from which they are derived. 

It may be stated that, excepting in cases where ex- 
treme accuracy is required, it makes little difference 
to the engineer as to which value he uses, nor from which 
steam table he procures any of the values used in steam 
calculations. As between the three latent heat constants 
given, a difference in results obtained will be less than 
1 per cent., w T hich is sufficiently close for all calculations 
in the realm of the practical operating engineer. 

In the formula, the quantity that changes 

970.4 

the actual evaporation of 1 lb. of water to equivalent 
evaporation from and at 212 F. is known as the factor of 
evaporation. In the example given, the factor of evapora- 
tion is: 

110=; — 200+32 

-^ ~ LA - = 1-0583 

970.4 D ° 

and when multiplying the actual total evaporation by 
the factor, the equivalent evaporation is obtained thus: 
3000X1.0583 =3174.90 which, as before given, would be 
called 3175 lb. in round numbers. 

The factor for any other values of steam pressure and 
feed-water temperature will be obtained and used in the 
same way as just illustrated. 

Boiler Efficiency 

The efficiency of a boiler plant is the ratio of the dif- 
ference between the heat in the steam (delivered by the 
boiler) and the heat in the feed water, to the heat that 



BOILER EFFICIENCY 71 

would be developed by the perfect combustion of the 
fuel. 

The following example illustrates the foregoing 
definition: 

The heat of combustion of a certain fuel is known to 
be 14,000 B.t.u. pe" pound; the number of pounds of 
such fuel used at a boiler test was 3500 per hour; the 
evaporation was 30,000 lb. of feed water per hour, into 
steam of 90 lb. gage pressure, or 105 lb. absolute; the 
temperature of the feed water was 8o° F. It is required 
to find the efficiency of the boiler. 

The method of operation is as follows: 

The total heat in 1 lb. of steam at 105 lb. absolute is 
1 187.2 B.t.u. (from Marks and Davis's tables). 

Temperature of feed w r ater = 8o° F., and 80 — 32=48 
assumed B.t.u. contained in the feed water above 32 F. 
Then: 1187.2—48 = 1139.2 B.t.u. and 1139.2X30,000 = 
34,176,000 B.t.u. absorbed by the water. 

As each pound of coal contains 14,000 B.t.u. and as 
the total coal consumed was 3500 lb. the total heat 
supplied = 14,000X3500 = 49,000,000 B.t.u. 

The efficiency is: 

34176000 X 100 



49000000 



= 69.74 per cent. 



A Short Method to Find the Commercial Effi- 
ciency of Boiler and Furnace Combined 

Expressed in terms of cost of evaporating 1000 lb. 
of water from and at 21 2 F. 

C 
C ~2Xe 



72 ARITHMETIC OF THE STEAM BOILER 

In which the values are as follows : 

c = cost of evaporating 1000 lb. t)f water from and at 

212° F. 

C = cost of coal per ton of 2000 lb. 

e = ihe evaporation per pound of coal, from and at 

212° F. 

2=a constant, to reduce the values to a common 
basis of 1000. 

Example, illustrating the formula: 

A certain coal costs $2.00 per ton of 2000 lb., and is known 
to actually evaporate 8 lb. of water per pound of coal consumed 
on the grates. The factor of evaporation — from the method 
previously explained — is found to be 1.0583. Therefore the equiva- 
lent evaporation from and at 212 F. is 1.0583X8 = 8.4664 lb. of 
water per pound of coal. 

2.00 
Then, c = wo — tt~ =S.ii8: or 11 8/10 cents per lb. 
' 2X8.4664 ' ' * 

As a check on the accuracy of the foregoing proceed thus: 

200 
Coal at $2.00 per 2000 lb., costs per pound, =.10 cents. 

Each 8.4646 lb. of water evaporated from and at 212 F. costs .1 
or 1/10 of a cent. As the cost for evaporating 1000 lb. is required, 
then 

1000 
~ 7—7 = 118.1 lb. of coal required 

and 1 18. 1 X.i =11.81 cents, as found before. 

How to Analyze a Boiler Trial Report 

The following is the report of a boiler trial which was 
held to determine the efficiency under given conditions. 



BOILER TRIAL 



73 



Sur- 



Pres- 



I 



i 



Tem- 
pera- 
tures. 



TRIAL OF A iooo H.P. BOILER 

i. Kind of trial running start 

and stop. 

2. Duration of trial 24 hours. 

3. Grate surface 113.6 sq. ft. 

4. Total heating surface 10,000 sq. ft. 

faces. ] 5. Ratio of heating surface to grate sur- 
face 88.0 

6. Average pressure per square inch, 
gage 81.0 lb. 

7. Average atmospheric pressure per sq. 
in 14.84 lb. 

8. Average absolute pressure per sq. in., 95.84 lb. 

9. Force of draft (column of water) 21 in. 

10. Temperature of the external air 30 F. 

11. Temperature of the fire room 93 F. 

12. Temperature of the feed water before 
entering the boiler 37-3° F. 

13. Temperature of escaping gases after 
leaving the boiler 407 F. 

14. Temperature of the steam 324+ F. 

15. Moist coal consumed 76,687 lb. 

16. Moisture in the coal 4.12 per cent. 

17. Dry coal consumed 73,528 lb. 

18. Total dry refuse 8069 lb. 

19. Total dry refuse 10.97 per cent. 

1 20. Total combustible 65,459 lb. 

21. Dry coal consumed per hour 3064 lb. 

22. Combustible consumed per hour 2727 lb. 

23. Percentage of moisture in the steam. 0.6013 per cent. 

24. Number of B.t.u. in 1 lb. dry coal. . . 14,500. 

25. Number of B.t.u. in 1 lb. combustible 15,425. 
f 26. Heat absorbed by the boiler, per 

pound of steam generated H75-6 B.t.u. 

27. Total B.t.u. absorbed by the boiler. . 774,880,282. 

28. Heat units imparted to the boiler per 
pound of dry coal 10,538. 

I, 29. Heat units per pound of combustible 11,838. 



B.t.u. «j 



74 



ARITHMETIC OF THE STEAM BOILER 






Water 



o 

a 



o 



30. Efficiency of the boiler, based upon 

dry coal (approximately) 72.7 per cent. 

31. Efficiency of the boiler, based upon 
combustible 76.77 per cent. 

32. Factor of evaporation 1.22. 

2,3. Total water fed to boiler 663,124 lb. 

34. Water actually evaporated, corrected 

for quality of steam 659,136 lb. 

35. Equivalent water from and at 212 F. 

boiler only 804,146 lb. 

36. Equivalent water from and at 212 F. 

per hour, boiler only 33, 506 lb. 

37. Water actually evaporated per pound 

of dry coal 8.96 lb. 

3S. Water evaporated per pound of com- 
bustible 10.07 lb. 

39. Horse-power, basis 34 1/2 lb. from 
and at 212 F 971. 

40. Number of square feet heating sur- 
face per horse-power 10.3. 

41. Horse-power per square foot of grate 8.53. 

42. Builder's rating, horse-power 1000. 



In the following, some of the results are not exact as 
far as actual numerals are concerned; this is due to dis- 
regarding too small decimal values, and using round num- 
bers instead as far as consistent. For practical purposes 
the values are sufficiently close ; the object in view is to ex- 
plain and illustrate how the values are found, rather than 
to give an exhibition of exact arithmetical operations. 

The first two items are self-explanatory. 

The grate surface (item 3) is found by multiplying 
together the length and width in feet, which gives th,e 
area in square feet. The heating surface (item 4) is the 
sum total of all surfaces which are in contact with the 



BOILER TRIAL 75 

hot gases on one side and water on the other side. The 
method of finding the heating surface has been explained 
in a previous section. 

The ratio of heating to grate surface (item 5) is 
found by division thus : 10,000^113.6 = 88.0. That is, 
item 4 is to be divided by item 3. 

The pressure of the atmosphere is noted (item 7) 
as 14.84 lb. per square inch. The pressure of the atmos- 
phere is usually considered as 14.7 lb. per square inch, 
and in ordinary calculations as 15 lb. in round numbers, 
at sea level. As a matter of fact the pressure of the 
atmosphere is constantly changing as indicated by the 
barometer. The way 14.84 was found is this: the height 
of the barometer divided by 2.04 equals atmospheric 
pressure. The 2.04 factor is the height of mercury in 
inches that is equal to 1 lb. per square inch. 

As an example, the barometer indicates 29.5 in., and 
29.5^2.04=14.4 lb. per square inch. Applying this, 
14.84X2.04 = 30.2+ in., which was the barometer read- 
ing at the time of the trial. 

Absolute pressure is measured from a perfect vacuum, 
and is gage pressure plus atmospheric pressure, in this 
instance 81 + 14.84 = 95.84 lb., item 8. 

The force of draft in column of water is 0.21 in. A 
draft gage is used for the purpose, one (1) ourice pressure 
per square inch is equal to 1.73 in. of water as registered 
by the gage. The value 1.73 is found like this: 
34 ft. =408 in. 
408 in. = 14.7 lb. (at sea level). 
14.7 lb. = 235.2 oz. 

Therefore, 1 oz. pressure = of 408 in., or 

Oj' 



76 ARITHMETIC OF THE STEAM BOILER 

408-T- 235.2 = 1.73 in. of water as indicated by gage. 

As a matter of convenience, the atmospheric pressure 

at the time of the trial is considered as having been 14.7 

lb. instead of 14.84, and the draft pressure in ounces is 

121 r 

0.21-M. 73=0. 121 or of an ounce. 

'° 1000 

The temperature of the steam (item 14) is found from 
a steam table, of which there are several in use; the 
one in particular gave the value of 324 F. corresponding 
to the pressure. 

The number of pounds of moist coal consumed was 
76,687 as actually weighed. As the coal contained 4.12 
per cent, of moisture, 4.12 per cent, of the 76,687 must 
be subtracted, which gives as a remainder 73,528 lb. of 
dry coal consumed (item 17), thus: 76,687X4.12 = 
3159.50 and 76,687-3159.50 = 73,527.5, say 73,528 lb. 
in round numbers. 

The moisture per cent, in the coal (item 16) is found 
like this: Exactly 100 lb. of coal, from the pile to be 
used during the trial, is placed in a bag or a box, and 
subjected to a good heat, such as would obtain on top of 
the boiler in operation, until it is thoroughly dried out. 
The coal is again weighed, and in the case under con- 
sideration was found to be 95.88 lb.; then, 100 — 95.88 = 
4.12 lb. moisture was evaporated, and this is 4.12 per 
cent, of the original weight. 

The total dry refuse — the non-combustible part of 
the coal — was 8069 (item 18) which is: (item 19) 
8069-^73,528 = 10.97 per cent. The total combustible 
(item 20) was: 73,528 lb. dry coal minus 8069 dry 
refuse = 65,459 lb. combustible. 



BOILER TRIAL 77 

The total quantity of dry coal consumed was 73,528; 
for one hour the quantity was 73,528-^24 = 3064 lb. 
(item 21). 

The combustible for one hour is found in the same 
manner. 

The percentage of moisture in the steam (item 23) 
0.6013 was determined by the use of a calorimeter. 

The heat value of the coal was determined by an 
analysis and test in a laboratory equipped with apparatus 
for that and similar purposes. In item 24 it is given as 
14,500 B.t.u. per pound dry coal, and in item 25, 15,425 
B.t.u. per pound of combustible. 

In item 26 the heat absorbed by the boiler per pound 
of steam generated was found like this: 

By referring to a steam table, the latent heat of steam 
at 95 lb. absolute pressure — which the report records — 
is found to be 886.7 B.t.u., while the sensible heat is 
323.89 and 886.7+323.89 = 1210.59, from which is to be 
subtracted the given temperature of the feed water, 37.3 
F., which gives 1210.59 — 37.3 = 1173.29 B.t.u. This 
method is approximate only. The following is more 
nearly accurate. 

The total heat of steam above 32 F. at 95 lb. absolute 
pressure is 1 180.7, as found in the steam table used in 
connection with this particular test. The temperature 
of the feed water above 32 F. was 37.3 — 32 = 5.3, and 
1180.7 — 5.3 = 1175.4 B.t.u., which more nearly agrees 
with the item in the report. 

The difference of 2. 11 which exists between the two 
results is accounted for in this way : 

The temperature of steam at 95 lb. absolute pressure 



78 ARITHMETIC OF THE STEAM BOILER 

is 3 23. 89 ° a,s found from the table used at that time. 
But strictly speaking, calculations involving the steam 
tables are based on water from 32 F. and not from zero 
on the Fahrenheit scale, as is sometimes done. Ignoring 
this will give approximate results only. Then, 323.89 
— 32 = 291.89; but the heat of the liquid as found in 
the table is 294.0 and 294.0—291.89 = 2.11, the difference 
before referred to. This difference is due to degrees 
temperature and B.t.u. 's not being exactly the same in 
value. However, the difference in the range of the 
steam tables is so small, that for all practical purposes 
where extreme accuracy is not demanded, degrees tem- 
perature may be used instead of heat in the liquid 
values. It is only as a matter of convenience that 
degrees temperature and units of heat are sometimes 
considered synonymous. 

The total heat units absorbed by the boiler during 
the trial was 774,880,282 (item 27) found like this: 
As 1175.6 is the heat units per pound of steam gene- 
rated by the actual quantity of water evaporated (cor- 
rected for the quality of the steam) which appears as 
659,136, then 1175.6X659,136 = 774,880,282 total 
B.t.u. absorbed. 

The heat units imparted to the boiler per pound of 
dry coal are found to be 10,538, found like this: 

Total B.t.u. Dry coal 

consumed 

774,880,282^-73,528=10,538 B.t.u. 

per pound of dry coal (item 28). 

The heat units imparted per pound of combustible 



BOILER TRIAL 79 

is found in a similar manner, only using the pound 
combustible as a divisor thus: 

774,880,282-7-65,459 = 11,838 B.t.u. 

per pound of combustible (item 29). 

The efficiency of the boiler based upon dry coal 
(item 30) is found by dividing the heat units per pound 
of dry coal by the theoretical heating value, which is 
taken as 14,500 B.t.u. thus: 10, 538-r- 14, 500 = . 7267 
which is 72.67 per cent. 

The efficiency based upon the combustible is: 

B.t.u. per lb. Theoretical 

combustible value 

11,838 -f- 15,425 = .7675 

w T hich is 76.75 per cent, (item 31); the efficiency of the 
boiler is the ratio of the heat utilized to that supplied. 
The factor of evaporation (item 32), 1.22, was found 
like this: 

1181-5-3 

~ — ^—=1.22 

05-7 
in which 1181 = B.t.u. in steam at 95 lb. pressure absolute. 

9 65.7 = B.t.u. 
required to evaporate 1 lb. of water from and at 212 F. 

5-3 = (37-3-3 2 ) 

The factor of evaporation means, that for every pound 
of water actually evaporated under the prevailing con- 
ditions at the time of the trial, 1.22 or 1 22/100 lb. of 
water would have been evaporated had the feed-water 



80 ARITHMETIC OF THE STEAM BOILER 

temperature been 212 F. and had the pressure been that 
of the atmosphere. 

Item 33 gives the total number of pounds of water 
fed to the boiler during the trial, as found by actually 
weighing it. 

Item 34 is found by multiplying the total weight 
of water fed to the boiler by the percentage of moisture 
as found in the steam expressed decimally, and then 
subtracting that value from the original quantity, thus: 

663, 124X. 006013 =3988— lb. 

moisture in the steam during the whole test, and 663,124 
— 3988 = 659,136 lb. actually evaporated. 

The equivalent evaporation from and at 212 F. (item 
35) is found thus: actual evaporation times factor of 
evaporation, or 

659,136X1.22 = 804,146 lb. 

The equivalent evaporation per hour (item 36) is 
found thus: 

804,146-^24 = 33,506 lb. 

The quantity of water actually evaporated per pound 
of dry coal consumed is 8.96 lb. (item 37) as found from 
dividing thfe total' water actually evaporated by the 
total dry coal consumed. Or, 

659,136-^73,528 = 8.96 lb. 

The water actually evaporated per pound of com- 
bustible (item 38) is found in a similar way, thus: 

659,136^65,459=10.07 lb. 



BOILER TRIAL 81 

Take the 8.96 lb. water per pound of dry coal, and 
multiply it by the factor of evaporation 1.22 thus: 
8.96X1.22 = 10.94 lb. of water from and at 212 F. on 
the dry coal basis. 

So, also in relation to the combustible, 10.07X1.22 = 
12.29 lb. water from and at 2i2°F. on the combustible 
basis. 

The horse-power is found (item 39) by dividing the 
pounds of water evaporated per hour from and at 212 
F. by the standard 34.5 thus: 33,506^34.5 = 971 H.P. 
The builder's rating was given as 1000 H.P. 

This is in excess of the actual power developed at the 
trial by 1000 — 971 = 29 H.P. and this expressed as a per- 
centage is 2.9 found thus: 

29 -J- 1000 = .029 or 2.9 per cent. 

By dividing the 10,000 sq. ft. heating surface by 971, 
the quotient obtained is 10.3 (nearly) which is the number 
of square feet per horse-power (item 40). 

And 1 13.6 sq. ft. grate surface divided into 971 gives 
8.53 H.P. per square foot of grate (item 41). 



PART II 

MISCELLANEOUS APPLICATIONS OF BOILER 
ARITHMETIC 



MISCELLANEOUS APPLICATIONS 

Bursting Pressure of Pipe 

2tXS 



P = 



D 



In which, P = bursting pressure, pounds per square inch. 
t = thickness in inches. 
S = tensile strength of the metal in pounds per 

square inch. 
D = internal diameter of pipe in inches. 

Example. — Find the bursting pressure of a io-in pipe, 0.366 in. 
thick, actual internal diameter 10.019 in. tensile strength of the 
metal taken as 50,000 lb. 

2 X. 366X50,000 

=3653 lb. 

10.019 ° °° 

For the safe working pressure, a factor of safety of 

not less than 10 should be used and preferably more. 

3653 
Therefore = 365.3 lb. safe working pressure. 

The bursting pressure being given, to find the thickness- 
of metal the formula is transposed thus: 

DXP 

85 



86 



ARITHMETIC OF THE STEAM BOILER 



Using the terms of the same question the statement 
becomes: 

10.019X3653 



t = 



2 X 50000 



= 0.366 in. 



When the exact tensile strength is not known, assume 
50,000 lb. for steel, and 40,000 lb. for iron pipe. 

The actual bursting pressure of pipes, as found from 
tests, is less than that found from the foregoing formula, 
and this makes it necessary that a liberal factor of safety 
be used. 



The Force Tending to Tear Asunder 
The force tending to tear asunder two cylinders of 




Fig. 14. — Lower part of Manning boiler. Diagram showing 
area against which the pressure acts, tending to tear apart the two 
cylinders. 

different diameters as illustrated in Fig. 14, which repre- 
sents a portion of the Manning boiler, the pressure 
being applied in the annular space between the two, 



MISCELLANEOUS APPLICATIONS 87 

depends upon the cross-sectional area of the space, and is 
independent of the diameter. The force tending to break 
the plates is the area of the sheets in the cross-hatched 
portion multiplied by the pressure per square inch. This 
force is resisted by the four plates, the outer and inner 
plates being in tension. 

The foregoing is based on the condition that the two 
cylinders are stayed to each other as shown in the 
illustration. 

Method of Finding the Fuel Cost of Evaporating 

1000 Lb. of Water 

data 

Assume that in a given case, 21 tons of coal were 
consumed in a certain time, and that the cost of the coal 
was $3.00 per ton of 2000 lb. 

Also assume that in the same period of time, 398,616 
lb. of water were evaporated into steam at a pressure of 
100 lb. per square inch absolute, the temperature of the 
feed w T ater being 200 F. 

The number of heat units in a pound of water at 200 
F. is 168.713 (as found from steam tables). 

The number of heat units in 1 lb. of steam at 100 lb. 
pressure per square inch absolute is 1181.866 (as fcund 
in tables). 

The number of heat units required to convert 1 lb. 
of water into steam at atmospheric pressure, 14.7 lb. 
per square inch, is 966 (in round numbers), and this is 
the latent heat of steam at 14.7 lb. pressure per square 
inch. 



88 ARITHMETIC OF THE STEAM BOILER 

METHOD OF OPERATION 

First, find the factor of equivalent evaporation to 

reduce the problem to the standard basis of, from and 

at 2i2° F. Employing the given values: 

1181. 866-168.7 

— -7-7— —=1.048 factcr of evaporation. 

The statement of the problem becomes: 

Tons Cents Lb. water 

21 X 300 X IOOO 



398616 X 1.048 

lb. water. Factor of 

evap. 



= 14.6 cts. 



or, $0,146, cost of evaporating 100c lb. water under the 
assumed conditions. The actual cost will be more than 
that found above, for labor, depreciation, cost of water, 
taxes, insurance, and interest on investment of the 
plant as a whole, enters the calculation where precision 
is required and over all charges are to be made. 

Safe Working Pressure for Cylindrical Cast-iron 
Vessels with Flat Cast-iron Heads 

rules prescribed by the united states inspectors of 
steam boilers 

When evaporators, feed-water heaters and separators 
are made of good cast iron, the shells cylindrical and the 
ends flat, the castings sound and of uniform thickness, 
the working pressure shall not exceed that found by the 
following formulas: For finding the safe pressure on the 
flat surface this is the formula to be used: 

_ 20000 XT 2 



MISCELLANEOUS APPLICATIONS 89 

For the cylindrical part of the vessel this is the formula 
to be used: 

3500(7^-1/4) 
r D 

And to find the thickness of metal required, having the 
other values given, use these formulas: 
For the flat heads 



-4 



TXD 2 



20000 
and for the cylindrical shell 

m PXD 



35°° 



1/4. 



In the formulas given the value of the letters stand like 
this: 

P = safe working pressure in pounds per square inch. 

T = thickness of metal in inches, provided the 
thickness of the ends or heads of such vessels 
shall not be less than 3/8 in. 

D = inside diameter of the vessel in inches. When 
the ends or heads are bolted to the shell then 
D = the diameter of the bolt circle. When the 
pressure is to be determined for a part of a 
flat surface which is square or rectangular, 
the value of D in the flat surface formula 
shall be the diagonal of the square or rec- 
tangle. The numbers 20,000 and 3500 are con- 
stants, evidently empirical, and found from 
experiment. 



go ARITHMETIC OF THE STEAM BOILER 

Equivalent Boiler Performance 

A boiler is sold on a guarantee that it will evaporate 
12 lb. of water per pound of combustible, from and at 
2i2° F., with coal having a heat value of 14,500 B.t.u. 
per pound of coal, and 8 per cent. ash. 

When the test was made the average feed-water tem- 
perature was 180 F., average steam pressure 70 lb., 
heat value of the coal used 12,000 B.t.u. per pound, and 
the ash content 8 per cent.; water evaporated, 9 lb. per 
pound of coal consumed. 

It is required to ascertain how the test compares with 
the guarantee. 

Under the conditions of the guarantee the coal is .08 
ash, and 1 — . 08 = .92 combustible. 

If 1 lb. of coal contains 14,500 B.t.u., 1 lb. of com- 
bustible will contain — ) — -=1^,760 B.t.u. 
.92 J " 

With this it is agreed to evaporate 12 lb. of water 
from and at 212 F. or 1 lb. under those conditions for 

. 15760 _ _. _ 

each = I 3 I 3 B.t.u. supplied. 

The temperature corresponding to 85 lb. absolute 
(70 lb. gage in round numbers) is 316.3 F. 

To raise 1 lb. of water from 32 to 316. 3 requires 
286.30 B.t.u. 

To raise 1 lb. of water from 32 to 180 requires 
147.88 B.t.u. 

To raise 1 lb. of water from 180 to 316. 3 requires 
138.42 B.t.u. 

To evaporate 1 lb. of water from and at 316.3 requires 
897.10 B.t.u. 



MISCELLANEOUS APPLICATIONS 



91 



To raise 1 lb. of water from 180 to 316. 3 , and evapo- 
rate it at 85 lb. pressure absolute, requires 1035.52 B.t.u. 
The coal used at the test contained 12,000 B.t.u. per 

r i 12000 
pound, and as there was 8 per cent, of ash, = 

13,043 B.t.u. per lb. of combustible. 

Under the conditions of the test, 1035.52 B.t.u. were 
required per pound of water evaporated into steam. 

13043 



Then, 



- = 12.59 lb. water evaporated per pound of 



io3S-5 2 
combustible, as compared with the guarantee of 12 lb. 

of water per pound of combustible. 



Efficiency of Diagonal Seam 
Assume a seam such as that shown in Fig. 15; the shell 




Fig. 15. — Diagonal seam. 

is 60 in. diameter, made of 5/16-in. plate having a tensile 
strength of 60,000 lb. per square inch. The seam is 20 



92 ARITHMETIC OF THE STEAM BOILER 

in. wide at one end and 6 in. at the other. The rivet holes 
are 11/16 in. in diameter and the pitch is 2 in.; the rivets 
are of steel. Assuming the efficiency of the joint to be 
41 per cent., what increase in efficiency is obtained by 
the seam being slightly diagonal to the axis of the shell? 
To find this in a simple manner, multiply the length of 
the seam in inches by 41, the efficiency of the joint, and 
divide the product by 60, which is the length in inches 
measured along a line parallel to the axis of the cylinder. 
Thus: As the seam, measured along the rivet center 
of the diagonal seam is 60.4 in., then, 

60.4X41 

— 7 = 41.27 per cent. 

efficiency of the diagonal seam, a gain of .27 per cent. 

Collapsing Strength of Cone-shaped Flue 

A cone-shaped flue has a greatest diameter of 36 in. 
and a least diameter of 12 in. and a length of 20 in. 
It is made of 5/16-in. steel plate of 60,000 lb. tensile 
strength per square inch. It is required to find what 
collapsing pressure such a flue will safely withstand. 
The construction is shown in Fig. 16. 

It is customary in short cones to take the mean 

diameter, in this case, 

36 + 12 
= 24 in., 

2 

and calculate the collapsing strength as follows: 
Hutton's rule is, 

T 2 XC _ 

dxVl~ p 



MISCELLANEOUS APPLICATIONS 



93 



The values of the letters are: 

T = Thickness of plate in thirty-seconds of an 

inch 
D = External diameter of the shell in inches. 
Z,=Length of shell in inches. 
C = 660 for mild steel plates. 
P — Collapsing pressure. 
Applying the formula, the statement becomes: 




Fig. 16. — Cone-shaped flue. 

10X10X660 
P= „ A w / — =614.0 lb. 
24XV20 

The collapsing pressure must be divided by the factor 
of safety, and in view of high temperatures and wear 
and tear, this should be 6; the allowable pressure will 
614.9 



be, 



; 102.5 lb. per square inch. 



The cone must be truly circular in form, and the brac- 
ing should be placed as shown in the figure in order to 
prevent the flue from being pushed down by the pressure 
exerted on the inclined surface. 



94 



ARITHMETIC OF THE STEAM BOILER 



Strength of Cone Seam 

A tank is 48 in. in diameter and built of 1/4-in. plate 
which has a tensile strength of 60,000 lb. per square inch. 
At the lower end of the tank is a cone w T hich has a single- 
riveted lap seam. The rivet holes are 11/16 in. in diam- 




Fig. 17. — Strength of cone seam. 

eter and the pitch is 2 in. What is the strength of the 
seam? 

Fig. 17 shows the construction and gives the 
dimensions. 

The longitudinal joint of a cone-shaped section which 
is withstanding internal pressure is subjected to a vary- 



MISCELLANEOUS APPLICATIONS 95 

ing stress. The stress at any point in such a joint will 
be inversely proportional to the distance of the point 
from the axis of the cone. 

It is not practicable to make a joint that would offer 
equivalent resistance to these varying stresses; therefore 
it is customary, when the cone is made of one sheet, to 
make it of the same strength as the tank itself. If 
several sheets are used in making up the length of the 
cone, the longitudinal joint of each course might be 
designed for strength inversely as its distance from the 
axis. This, however, is rarely done except in large tank 
work where the size makes the saving in material an 
object. 

The strength of the seam is the strength of the weakest 
part, which, neglecting the crushing of the sheet in 
front of the rivets must be either the tensile strength of 
the ligament between the rivets, or the shearing strength 
of the rivets themselves. 

The strength of the portion of the plate between the 
rivets is: 

2 — .6875 60000 __ . , _ 
X =9943.75 lb. per inch of seam length. 

Steel rivets in single shear are assumed to have a 
shearing strength of 42,000 lb. per square inch of cross- 
sectional area; the cross-sectional area of a rivet n/16 in. 
in diameter is .37122 sq. in., and the strength of the 
rivets is 

.37122X42000 r „ 
— -=7795-62 lb. 



96 arithmetic of the steam boiler 

per inch of seam length, which, being weaker than the 
ligament, is the strength of the seam. 

While the seam is of equal strength throughout its 
length, the internal pressure per square inch required to 
shear the rivets will vary directly as the diameter of 
the cone. For instance, on a line corresponding to a 
diameter of 40 in. the bursting pressure of the cone 
will be 

*£*-*«« >b. 

per square inch. For a diameter of 16 in. the seam will 
fail at 



7705.62 

^^| — = 974.45 lb. 



per square inch. 



Collapsing Pressure of Fire Box 

Fig. 18 illustrates a vertical fire-box boiler. It is 
required to figure the collapsing pressure of the fire box, 
assuming that 7/8-in. stay bolts are used and that the 
pressure is 125 lb. per square inch; it is also required to 
find what pitch to give the rivets in the vertical seam, 
using 11/16-in. rivets. 

In calculating the strength of cylindrical furnaces 
supported by stay bolts, it is customary to assume that 
the surface is flat; that is, the tendency of the form to 
lend strength to the construction is ignored. 

With a specified size of bolt the first step is to find. how 



MISCELLANEOUS APPLICATIONS 



97 



many square inches of surface one bolt will support at 
the given pressure of 125 lb. per square inch. 

Standard stay bolts up to 
1 1/4 in. and larger are cut 
12 threads per inch, and it is 
practically correct to assume 
that the depth of thread is 
the same as given by the 
United States Standard for 
that pitch, viz., .05425 in., 
. although this is not strictly 
correct, as the usual stay- 
bolt thread varies slightly 
from this standard. 

Using 7/8-in. bolts will 
give an effective area of 
.4614 sq. in. per bolt. The 
working stress allowed in 
stay bolts varies, according 
to different authorities, from 
6000 to 10,000 lb. per square 
inch, the most generally ac- 
cepted figure being 7500, and 
on this basis a 7/8-in. stay 
would support 3460 lb. or 
at a pressure of 125 lb. per 
square inch it would be capa- 
ble of supporting 
3460 




r 



^—2 hV 2 V 

— T.S. 
56,000 Lbs. 



/ 



I 



I 2 S 



= 27.6 



Fig. 18. — Diagram from 
which to calculate the collaps- 
ing pressure of fire box. 



square inches of surface, or the bolts could be spaced 



98 ARITHMETIC OF THE STEAM BOILER 

\/27.6, or about 5 1/4X5 1/4 in. Of course to be 
strictly correct, the area occupied by the bolt itself 
should be added to the 27.6 in. before extracting the 
square root to get the pitch, for the pressure does not 
act on this area. But in actual practice such allowance 
is not made. 

The surface to be supported in the problem is 2 ft. 
6 1/8 in. in diameter, or 94.64 in. in circumference, 
and the approximate height or distance between the rivets 
on the leg ring and crown sheet is 24 in. and with the 
spacing given before it would be necessary to have 18 
vertical rows of stays containing four stays each. 

There is another feature to be considered in the stay- 
ing of sheets that affects the strength, and that is the 
stresses in the sheets themselves. Unwin has devised a 
formula for the maximum stress in flat sheets supported 
at regular intervals, the supported points forming squares. 
The formula is: 




in which a is a side of a square, / the stress per square 
inch in the sheet, / the thickness of the sheet in inches, 
and p the pressure per square inch on the supported 
surface. 

Substituting present values in this formula, and assum- 
ing a maximum working stress in the sheet of 7500 lb. 
per square inch, the statement becomes: 



o= V 



9x7500x.3125x.3125 
2x125 5 3> 



MISCELLANEOUS APPLICATIONS 99 

or 5 1/8 in., which is a little less than the pitch deter- 
mined by the strength of the stay bolt. The first lay- 
out should be modified, making 19 vertical rows of stays 
with four stays in each row, or the pitch would be about 
5X4.8 in. 

In relation to pitching the rivets in the vertical seam, 
it is found in practice that the lapping together of the 
plate at this point stiffens the sheet so that strength of 
the joint is not a very important factor in the case, 
and this joint is generally designed to be least affected 
by the heat, i.e., single riveted, and of the best propor- 
tions to insure tightness, which w T ould be a pitch of about 
2 1/4 in. if 5/16-in. plate and n/16-in. rivets be used. 

Relating to Safety-valve Rules 

Concerning the apparent discrepancy between the 
different rules relating to safety-valve calculations, 
particularly that in Reed's Engineer's Handbook (an 
English publication), which is used by candidates for 
British Board of Trade certificates of competency, and 
those used by a prominent educational institution the 
reader w T ill discover, after carefully reading the following 
statements, that both rules referred to will give the same 
results in any given problem when intelligently used. 
In other words, both rules are correct, although there 
is a possibility of misunderstanding them and so obtain- 
ing incorrect answers. 

If the safety-valve problem could be handled without 

taking into consideration the effective weight of the lever, 

then it is not probable that there would ever be any 

difficulty in candidates having any misunderstanding 

8 



IOO ARITHMETIC OF THE STEAM BOILER 

of the matter. But the effective weight of the lever 
must be taken into the calculation if accuracy is desired, 
and of course in such a matter accuracy should be 
desired. Consider what the effective weight of a safety- 
valve lever is and what relation it bears to the calculation. 
In the first place there is a difference between the 
effective weight of a lever and the effective moment of a 
lever. Either may be used in the safety-valve problem, 
but it must be clearly understood how and where each 
is to be used. Probably the whole difficulty that candi- 

, L =20 



10 " 



F = 2' 



H 



..4" Fva| 



Center of Gravity of Lever 

f W? 




Dia. 4 1/ V \]P= 1256.64 lb. 

Fig. 19. — Diagram for safety-valve calculations. 

dates have with the safety-valve problem lies just in 
this point. The effective weight of a lever is found by 
multiplying its weight in pounds by the distance its center 
of gravity is from that point called the fulcrum, and by 
dividing the product by the distance that the center 
of the valve upon which the lever acts is from the ful- 
crum. The effective moment of a lever is found by multi- 
plying its weight in pounds by the distance its center of 
gravity is from the fulcrum. The distances referred to 
are measured in inches. 

Referring to the following example: Required, the 
weight to be placed at the end of the safety-valve lever 
shown in the diagram with the given data. 



MISCELLANEOUS APPLICATIONS ioi 

Diameter of valve, 4 in. 

Area of valve, 12.5664 sq. in. 

Steam pressure per square inch, 100 lb. 

Weight of lever, 20 lb. 

Weight of valve and stem, 10 lb. 

Total upward pressure on the valve is 1256.64 lb. 

The letter c represents the sum of the effective moments 

of the valve and stem, and also the lever. The effective 

moment of the lever is 10X 20= 200; the effective moment 

of the valve is 2X10=20; and the sum is 200+20 = 220. 

xt FP-c . (2X1256.64)^-220 

Now, — y — = weight, or -=114.66410., 

the weight to place at the end of the lever. (This 
method employs effective moments of valve and lever.) 

Now try the same example by using the other rule, which 
employs the effective weight of the lever, and the actual 
weight of the valve. The rule is stated thus: "To 
find the weight which must act on a lever at a given dis- 
tance from the fulcrum so that the valve is about to blow 
off at a given pressure, subtract the downward force due 
to the weight of the valve, stem and lever from the prod- 
uct of the area and the steam pressure. Multiply the 
remainder by the distance from the fulcrum to the 
center line of the valve, and divide this product by the 
distance from the fulcrum at which the weight is to act." 
Expressed in a formula it appears like this: 

w J Ap -^ D 

W L 

W = the required weight in pounds. 

A = the area of the valve in square inches. 



102 ARITHMETIC OF THE STEAM BOILER 

P = the pressure per square inch. 

D = distance in inches from the fulcrum to the 

center line of valve. 
L = distance from fulcrum at which the weight 

is to be placed. 
w = the weight of the valve and stem in pounds 

plus the effective weight of the lever. 
Weight of valve and stem = 10 lb. 

Effective weight of lever = — = ioo lb. 

° 2 

Total downward force due to both valve and lever = 
no lb.; that is, w in the formula = no lb. 

( 1 2 . 5 664 X i oo — 1 1 o) X 2 

— = 114.6^4 lb., 

20 J 

the same answer as obtained by the first method 
employed. 

The effective weight of the lever will be = 100 

lb.; that is, this lever, because of its weight and its point 
of contact with the valve, is equivalent to 100 lb. being 
placed directly on top of the valve. 

The effective moment of the same lever will be: 20 X 
10= 200. 

Now, either the effective weight of the lever or the 
effective moment may be used in the calculation as has 
been shown, but each has its own place in the operation, 
as will be seen a little further on. 

The equality of moments in the safety-valve problem 
are stated like this: 

WXD+W'XD'+wXd=pXAXd 



MISCELLANEOUS APPLICATIONS 103 

In which 

W = weight of the weight in pounds. 

D = distance in inches that weight is placed from 
fulcrum. 

W' = weight of the lever in pounds. 

D f = distance of center of gravity of lever from 
fulcrum. 

w = weight of the valve and stem in pounds. 

d = distance center of valve is from the fulcrum. 

p = pressure in pounds per square inch. 

A = area of valve in square inches. 
By applying the formula to the problem it can be seen 
that the answer obtained is correct. Showing the let- 
ters, and substituting the numerals of the example, 
the statement becomes: 

WXD+W'XD'+wXd=pXAXd, 

114.664X20+20X10+10X2 = 100 X 12.5664X2, 

229.328+200+20 = 2513. 28, 
2513.28 = 2513.28. 

Any safety-valve rule or formula that can successfully 
stand the above test is correct and may be safely 
employed. 

In order to arrive at the point aimed at, and also to 
make it as clear as possible, it may be said that from the 
foregoing formula of moments a series of formulas can be 
derived that will handle the safety-valve problem in all 
its phases. Here are the formulas: 

(1) FX S+ L = W. 

(2) FX S + W=L. 



104 ARITHMETIC OF THE STEAM BOILER 

(3) LXW+ S=F. 

(4) LXW+ F = S. 

Where F = the force acting upward against the valve; 
this equals the pressure per square inch times 
the area of the valve; from the product must 
be subtracted the weight of the valve and 
stem. 
5 = the distance from the fulcrum to the center 

of the valve. 
L = distance from the fulcrum to point where the 
w T eight is hung. (This may or may not be at 
the extreme end of lever.) 
W =the weight in pounds of the weight to be hung 
on lever. 
In cases (1) and (2) the effective weight of the lever 
must be subtracted just before multiplication occurs. If 
effective moment of lever is to be employed, then the 
effective moment is to be subtracted just before division 
occurs. 

In cases (3) and (4) the effective weight of lever is to be 
added just before multiplication occurs. 

If the effective moment of lever is to be employed, then 
it will be added just before division occurs in the formula. 
It is very important that a distinction be made between 
the two phases of the four cases, and then no trouble will 
be experienced. 

Bearing in mind what has just been stated and being 
ready to refer to it again, now try the problem by using 
Reed's rule, which is stated thus: 

(1) Find the area of the valve and multiply it by the 
pressure per square inch. 



MISCELLANEOUS APPLICATIONS 105 

(2) From the product take the weight of the valve 
(which of course includes the stem). 

(3) Multiply the remainder by the distance from the 
fulcrum to the valve, then subtract the moment of the 
lever, and divide by the distance from the fulcrum to the 
weight. 

In Reed's Engineers' Handbook the effective moment 
of the lever is defined the same as that which appears 
in the earlier part of this subject. 

Using the figures in the example and following Reed's 
rule, 12.5664 sq. in. (area of valve) X 100 (pounds pressure 
per square inch) = 1256.64 lb. (total upward pressure) 

— 10 lb. (weight of valve and stem) = 1246.64 lb.X2 
(inches distance from fulcrum to valve) = 2493.28 lb. 

— 200 (moment of lever) = 2293.28 lb.-^2o (inches, 
distance fulcrum to weight) = 114.664 lb. weight. Ans. 

Thus it is seen that exactly the same answer is obtained 
as found before. Reed's rule is therefore correct if 
intelligently used. 

Now, suppose that it is desired to use Reed's rule, but 
instead of having the effective moment of lever given, the 
effective weight is given instead. The effective weight 

of the same lever would be (as before explained) 

= 100 lb. The operation will be like this: 12.5664 sq. 
in. (area of valve) X 100 (pounds pressure per square 
inch) = 1256.64 lb. (total upward pressure against valve) 

— 10 lb. (weight of valve and stem) = 1246.64 lb. 
(balance) — 100 lb. (effective weight of lever) = 1146.64 
lb. X2 (inches, distance, fulcrum to valve) = 2 293. 28 4- 20 
(inches, distance, fulcrum to weight) = 114.664 lb. weight 



106 ARITHMETIC OF THE STEAM BOILER 

answer, giving exactly the same answer as in the other 
two cases. Again is attention directed to the difference 
between the tw r o latter cases, and particularly the place 
in each w T here the subtraction of the lever factor takes 
place. 

Suppose the effective weight of the lever were subtracted 
after multiplication had occurred instead of before, then 
the following statement would occur: 12.5664X100 = 
1256.64—10=1 246.64 X 2 = 2493 .28 — 100 (effective weight 
of lever) = 2393.28-^20=119.664 lb. w r eight. 

119.664—114.664 = 5 lb. difference, too much weight. 
In other words, by so doing the valve would be over- 
weighted 5 lb. Of course, the percentage of difference is 
small, but it is further aggravated by the frictional 
resistances, and it is not on the safe side of the calcu- 
lation. 

Suppose on the other hand the effective moment of the 
lever were subtracted before multiplication occurs; the 
statement would be: 12.5664X100=1256.64—10 = 
1246.64—200 (effective moment of lever) = 1046.64-^20 
= 104.664 lb. weight. 

114.664—104.664=10 lb. difference, too small. 

Let the interested reader study this matter out for 
himself and try the arithmetical operations in the 
different cases. In this way he cannot but come to a 
correct and complete understanding of all that is em- 
braced in the problem, as far as an operating engineer is 
concerned, when standing an examination for a certificate. 

Any rule relating to the safety valve, which a candidate 
is not sure of should be tested by the application of the 
equality of moments formula before referred to. 



MISCELLANEOUS APPLICATIONS 107 

Roper's Safety-valve Rules 

Examiners of engineers in the United States Steam- 
boat Inspection Service sometimes prefer to have candi- 
dates for American marine engineers' license use what 
are known as Roper's Rules for safety-valve problems. 
Therefore it is well to have a knowledge of these rules 
in case they are required. 

In the following formulas let 

A = Area of valve in square inches, or diameter 2 

X.7854. 
D = distance from center of valve to the fulcrum, 

in inches. 
L = distance of the weight from the fulcrum, in 

inches. 
P = steam pressure in pounds per square inch. 
W = weight of the ball in pounds to hang on the 

lever. 
V = w r eight of the valve and stem in pounds. 
w = w eight of the lever in pounds. 
/ = distance of the fulcrum from the center of 

gravity of the lever, in inches. 

n (WXL) + (wXl) + (VXD) 



AXD 



(1) 



TT7 AXPXD-{wXl+VXD) , x 

W = — £ — (2) 

■ AXPXD-(wXl+VXD) . , 

L= — w~ (3) 

The following examples illustrate the application of 
the formulas given. 



108 ARITHMETIC OF THE STEAM BOILER 

Example i. — At what pressure will a safety valve blow off, hav- 
ing a diameter of 4 in., weight of valve and stem 12 lb., weight of 
lever 22 lb., weight of ball 125 lb.; the overall length of lever is 46 
in., and it is straight and parallel; the weight is hung at 42 in. from 
the fulcrum, and the distance from center of valve to fulcrum is 
4 in. 

Using formula (1) the statement becomes: 



P = 



(125X42)+ ( 22 X^) +(12X4) 

4 2 X. 7854X4 
= 115.466+ lb. per square inch, answer. 

Example 2. — Using the foregoing example (1), it is required to 
find the necessary weight to hang on the lever. 
Using formula (2) the statement becomes: 



W 



4 2 X. 7854X115. 466X4- (^2Xy + I2 X4) 



42 
= 125 lb., answer. 



Example 3. — Using the same example (1), it is required to find 
at what distance the weight is to be hung. 
Using formula (3) the statement becomes: 

12. 5664X115. 466X4- ( 22X^+12X4) 



125 
=42 in., answer. 

Safety-valve Capacity 

Extract from paper read before the A. S. M. E., 
February 23, 1909, by Philip G. Darling, Mechanical 
Engineer. 



MISCELLANEOUS APPLICATIONS 109 

CAPACITY FORMULA FOR 45° SEATS 

1. E=iosXlXPXD 

2 . D = .oo 95 j£p 

Modified Forms for Special Applications 
For Locomotives 

H 

For Cylindrical Multitubular, Vertical and Water Tube 
Stationary Boilers 

For Water Tube Marine and Scotch Marine Boilers 
5. D = .o 9 5j^p 

E = pounds of steam relieved per hour. 
/ = vertical lift of valve in inches. 
P = steam pressure (absolute) pounds per square 

inch. 
D = nominal diameter of valve (inlet) in inches. 
H = total boiler heating surface in square feet. 

For flat-seated valves the constants in these formulae 
are as follows: i — 149.; 2 — .0067; 3 — .052; 4 — .065; 
5— .090. 



no ARITHMETIC OF THE STEAM BOILER 

UNITED STATES STEAMBOAT INSPECTION SERVICE SAFETY- 
VALVE RULE 

The rule is that the areas shall be found by the formula: 

WG 

A = .2o 74 x 1 r' 

in which .4 = the area of the safety valve in square 
inches. 
W= pounds of water evaporated per square 

foot of grate surface per hour. 
P = the absolute pressure per square inch. 
G = grate area in square feet. 
In the case of spring-loaded valves, the effective area 
must be equal to that derived from the formula, and a 
lever must be provided which will raise the valve one- 
eighth its diameter from its seat. All seats to have an 
angle of 45 to the axis of the valves. 

Derivation of the United States Board of Super- 
vising Inspectors' Rule for Areas of 
Safety Valves 

Napier's rule for flow of steam through orifices: 

_. . . . Absolute pressure X area 

Flow in pounds per second = 

(This corroborated by Peabody's experiments.) 
P = absolute pressure = gage pressure+ 1 5 . 
W = pounds discharged per hour. 
A =area of valve opening or orifice. 



MISCELLANEOUS APPLICATIONS in 

Hence 

_ PXA * e , c 'T 36oX^XP 
W= X6oX6o= — — 

70 7 

For safety-valve practice, cut this amount down 25 
per cent., leaving 75 per cent. 
Thus 

Restrict the lift of valve to 1/32 of its diameter = 

d_ 
then 32 

d ~Xd 2 

A — X^Xd = lift X circumference = — 

3 2 3 2 

Substituting this value for A = area of orifices 
270 „ 7ld 2 

w=^~xpx — 

7 32 

In a valve of diameter d the area = 

Tld 2 

=a 

4 

To get W in terms of area of valve, substitute for d 2 its 
value in terms of a, 

W =- 7 -X P X— X 4 - =4-821 XPa 

7 32 t 

In safety-valve practice this will represent the pounds 



112 ARITHMETIC OF THE STEAM BOILER 

of steam that must escape per hour, which must be equal 
to the pounds of water that the boiler can evaporate per 
hour. 

To reduce this to a working basis, consider these quanti- 
ties per square foot of grate surface per hour. 

W — pounds of water evaporated per square foot 

of grate surface per hour. 
P = absolute pressure per square inch. 
A =area of safety valve per square foot of grate 
surface. 
Hence 

W 
TF=4.82iXPX^, anda = .2074X -p 

From which a table of areas required per square foot of 
grate surface may be found by assuming the different 
values of W and P. 

Finding the Center of Gravity of Tapered Safety- 
valve Levers 

In questions relating to the lever safety valve it is 
necessary to know how to find the center of gravity of 
the lever, in order to calculate the effective weight of the 
lever; how may the center of gravity be found w T hen the 
lever is tapered, and of uniform thickness throughout its 
entire length? 

Answer. — There are three ways that the center of 
gravity of tapered safety-valve levers is found: By 
taking the lever off and actually balancing it on a knife 
edge; diagrammatically, and mathematically. The last 
two methods only require an explanation. 



MISCELLANEOUS APPLICATIONS 



113 



First consider the diagrammatic method of finding the 
center of gravity. To be brief and to the point, assume 
a lever 20 in. long, 2 in. wide at one end, and 1 in. wide at 
the other end and of uniform thickness throughout. 
Do not consider the projection which is usually at the 
small end of the lever for preventing the weight slipping 
off, nor the holes at the other end through which the 
lever is attached to the fulcrum, and by which the pin is 
attached which bears upon the valve. These would 




Fig. 20. — Diagram for finding center of gravity of tapered lever. 



make but little difference in the result obtained, and 
would tend to complicate the calculations. 

An inspection and study of figure 20 should make clear 
how the center of gravity may be determined. A scale 
of quarter size may be chosen as a matter of convenience 
only. Any other scale w T ill do. The center line AB is 
' drawn first. The length of A B is 5 in., or one-quarter as 
long as the lever assumed. At A and B respectively 
draw the lines CD and EF 1/2 in. and 1/4 in., represent- 
ing the actual 2-in. and i-in. dimensions of the lever. 

Next draw the diagonal line DE, and locate the point 1 
exactly midway between points D and E. Join the 



114 ARITHMETIC OF THE STEAM BOILER 

points C and i by the line Ci. From the point i locate 
the point 2 on the line Ci at a distance equal to one- 
third of its length. Now, from the point F draw the 
line Fi, and from the point 1 locate the point 3 on the 
line Fi, one-third of its length from the point 1. Join 
points 2 and 3; where the line 2-3 intersects the center 
line AB will be the point 4 at w T hich the center of gravity 
is. By carefully and accurately drawing the figure the 
measurements show that the center of gravity is about 
1 1. 1 in. from the small end of the lever, and is about 
8.9 in. from the large end of the lever. 

To find the center of gravity of a tapered lever mathe- 
matically, the following formula may be used: 

2A+B T . ,. p 2A+B 

X== 3 A+$B X ; ° r m m? x= siA + B) X 

Where x = distance center of gravity is from the small 
end of lever. 
A= the width in inches of lever at the fulcrum 

end. 
B = width in inches of lever at the weight end. 
L = entire length of lever in inches. 
Applying this formula to our example we have 

2X2 + 1 5 100 

-7 : rX20=~X20= "=11.11 111., 

3(2 + 1) 9 9 

distance center of gravity from small end of lever. 

Thus it will be seen that the two methods which have 
just been explained verify each other, and no doubt 
the first method (that of balancing the lever on a knife 
edge) referred to would verify the others, for, as before 



MISCELLANEOUS APPLICATIONS 115 

stated, the difference due to the projection at one end 
and the holes at the other end would hardly be noticeable. 
At an examination where time is limited, it would be 
better to use the formula in such questions. It does not 
require much study and only a little practice to become 
familiar with it. Furthermore, it will be found easy to 
remember after it has been used a few times. Marine 
engineers cannot afford to ignore this subject. 

Chimneys 

The " proportions of chimneys" vary very much 
according to the requirements. Every chimney should 
be large enough in cross-section to carry off the gases and 
high enough to produce sufficient draft to cause a rapid 
combustion. The object of a chimney being to carry 
off the waste gases, it naturally determines the amount 
of fuel that can be burnt per hour, and it is advisable to 
have always a good draft, as it can always be regulated 
by a damper. 

Draft pressure is caused by the difference in weight 
between a column of hot gases in the chimney and a 
column of air of equal height and area outside the 
chimney. 

Formula for finding the force of draft in inches of 
water of any given chimney: 

7-64 7-95^ 



^tj(7M 7-95\ 



Where F = force of draft in inches of water. 
H = height of chimney in feet. 
9 



Il6 ARITHMETIC OF THE STEAM BOILER 

T\ = absolute temperature of chimney gases 

(H-460). 
T 2 = absolute temperature of the external air 

(/1+460). 
t = temperature of chimney gases. 
ti = temperature of external air. 
Formula for finding the height of a chimney in feet for 
a given force of draft: 

F 



11 = 



/7.6 4 _7. 9 5\ 
\T 2 Tj 



To find the maximum force of draft for any given 
chimney, the external air being 6o° F., and the heated 
column being 6oo° F., multiply the height above the 
grate in feet by .0073, and the product is the force of 
draft expressed in inches of water. 

1. The draught power of the chimney varies as the 
square root of the height. 

2. The retarding of the ascending gases by friction 
may be considered as equivalent to a diminution of the 
area of the chimney, or to a lining of the chimney by a 
layer of gas which has no velocity. The thickness of 
this lining is assumed to be 2 in. for all chimneys, or 
the diminution of area equal to the perimeter X 2 in. 
(neglecting the overlapping of the corners of the lining). 
Let D = diameter in feet, A = area, and E = effective area 
in square feet. 

2 8D 2 — 

For square chimneys, E = D — =A — \/ A 

For round chimneys, £=- yD 2 — — j =A — .591 \A4 



MISCELLANEOUS APPLICATIONS 117 

For simplifying calculations, the coefficient of \/ A 
may be taken as .6 for both square and round chimneys, 
and the formula becomes 

3. The power varies directly as this effective area E. 

4. A chimney should be proportioned so as to be 
capable of giving sufficient draught to cause the boiler 
to develop much more than its rated power, in case of 
emergencies, or to cause the combustion of 5 lb. of fuel 
per rated horse-power of boiler per hour. 

5. The power of the chimney varying directly as the 
effective area E, and as the square root of the height 
H, the formula for horse-power of boiler for a given size 
of chimney will take the form horse-power = CE\/e[, in 
which C is a constant, the average value of which, ob- 
tained by plotting the results obtained from numerous 
examples in practice, the author finds to be 3.33. 

The formula for horse-power then is 

horse-power = 3. 3sE\/jj y or horse-power = 3.33 (A — 

aVaWh 

If the horse-power of boiler is given, to find the size 
of chimney, the height being assumed, 

.3 H.P. ,- 

For round chimneys, diameter of chimney = diam. of 
£+4 in. 

For square chimneys, side of chimney = \/£+4 in. 



Ii8 ARITHMETIC OF THE STEAM BOILER 

If effective area E is taken in square feet, the diameter 
in inches is d= i^.S4\/E+4. in., and the side of a square 
chimney in inches is s= i2\/E+4 in. 

If horse-power is given and area assumed the height 

In proportioning chimneys the height is generally 
first assumed, with due consideration to the heights of 
surrounding buildings or hills near to the proposed 
chimney, the length of horizontal flues, the character 
of coal to be used, etc., and then the diameter required 
for the assumed height and horse-power is calculated by 
the formula or taken from the table. 

Size of Boiler Feed Pipe 

What size feed pipe should be installed to supply 
three ioo-H.P. boilers? 

At ordinary commercial rating the water required per 
horse-power-hour, is taken as 30 lb. Each boiler there- 
fore will need 3000 lb. of water per hour or 50 lb. per 
minute, which at 62.5 lb. per cubic foot would call for 
.8 of a cubic foot each minute. 

To provide for emergencies, twice the actual quantity 
of water required should be figured. The velocity of 
flow is usually taken as 100 ft. per minute. On this 
basis, the quantity to be taken care of will be 2X.8 = 
1.6 cu. ft. per minute, which at 100 ft. per minute velocity 

would require an area of pipe of -=.016 sq. ft., or 

100 

2.3 sq. in., found like this: 

.016X144=2.304 sq. in. and A / '^ I *7 m - diam. 

\-7 8 54 



MISCELLANEOUS APPLICATIONS 1 19 

The nearest commercial size of pipe is 1 1/2 in. diameter, 
which is the size to connect to each of the boilers. 

For the main pipe, to supply all three boilers, a pipe 
of equivalent area to the three boiler pipes would be 6.9 
square inches, and this would mean a 3-in. diameter 
pipe is required. Should not all three boilers require 
to be fed at the rate before referred to at any one time, 
then a 2 1/2-in. main feed pipe will be ample. 

Effect of Stiffness of Head on Braces 

Away back in 1876, Samuel Nichols, a practical boiler 
maker in charge of a large English works, wrote a book 
for boiler makers. In it he recounts some experiments 
made under the supervision of Robert Nelson, author 
of " A Treatise on Steam Boilers/' upon boilers built by 
himself for the purpose, with regard to unstayed heads in 
cylindrical shells. A boiler 30 in. in diameter with flat 
heads 3/8 in. thick, of plate having a "tenacity" of 
21.2 tons per square inch with an elongation of 7.9 per 
cent., flanged on a radius of 1 in. inside the plate, was 
subjected to hydrostatic pressure. At 10 lb. the head 
had bulged 1/16, at 120 lb. 11/16 and at 150 lb. 13/16 in. 
Permanent set occurred somewhere between 50 and 65 
lb. at which latter figure the deflection was 3/8 in. 
Rupture took place at about 300 lb. 

From the results of his tests, Mr. Nichols deduced a 
formula which is printed in his book for determining 
the bursting pressure of cylinders with unstayed heads, 
although he strongly disclaimed any advocacy of ex- 
posing an unstayed surface to high pressures. "On the 



120 ARITHMETIC OF THE STEAM BOILER 

contrary," he writes of himself, "he is more convinced, 
now that he has witnessed these experiments, that a fiat 
unstayed surface is very weak indeed, and that they still 
require a larger amount of care and judgment on the 
part of boiler engineers than any portion of the boiler/' 

Evidently under the erroneous notion that if the head 
can take care of the amount of pressure indicated by this 
formula on its own account it will need correspondingly 
less bracing, the Board of Boiler Rules of Ohio, after 
instructing inspectors to determine the working pressure 
of boilers with respect to the bracing in the usual way 
but with an allowable stress of 8000 lb. per square inch 
irrespective of size, tells them, "To the above pressure 
may be added the Nichols formula w T ith a factor of 
safety of not less than 8." 

If this means, and it can seem to mean nothing else, 
that to the pressure which can safely be taken care of by 
the bracing may be added the pressure which, by the 
Nichols formula, would be allowed upon the unbraced 
head, it is wrong. 

Mr. Nichols shows that the head may be bulged 
considerably without straining the sheet beyond the 
elastic limit. But a brace is supposed to be tight before 
the head commences to bulge. Just as soon as the head 
starts to move it commences to stretch the brace. 
Under the allowable stress of 8000 lb. per square inch of 
section a brace wall extend only about 1/3750 of its length, 
or a 6-ft. brace would extend less than .02 in. The pres- 
sure which would produce the movement in the unbraced 
head is inconsiderable — a pressure of 10 lb. produced a 
movement of three times as much — and yet this is 



MISCELLANEOUS APPLICATIONS 121 

all the help that the stiffness of the head would be to the 
brace. 

As an example, assume a 72-in. boiler, height of seg- 
ment to be braced 24 in., area of segment to be braced 
814 sq. in., pressure 100 lb., thickness of head 1/2 in., 
tensile strength 60,000 lb. 

The Nichols formula with a factor of safety of 8 
would allow 

tXT Xio 0.5X6000 0X10 
^4X8 "814X8 -46 1b. 

If this may be added to the pressure which the bracing 
is capable of carrying, it would be necessary to brace 
against only 100 — 46 = 54 lb. per square inch, which at 
8000 lb. per square inch w^ould require seven i-in. braces. 
Practice calls for at least twice as many. 

Bracing Flat Surfaces in Steam Boilers 

There is considerable variation as to the load allowed 
per square inch of net section on diagonal braces, rod 
braces and stay bolts by the authorities who have 
laid down rules on this subject. The United States 
Government rules allow 6oco lb. on welded iron stays 
below 1 1/4 in., 7500 on 1 1/4 in. and above, and from 
7000 to 9000 lb. on weldless steel stays. Chicago has a 
flat scale of 6000 lb. on all stays or braces, Philadelphia 
has a limit of 7500 lb.; the Massachusetts rules allow 
from 6500 to 9000 lb. per square inch net section, varying 
as the braces are w r elded or weldless and with the size, 
the latter for the reason that with a given waste of 



122 ARITHMETIC OF THE STEAM BOILER 

material the percentage of reduction is greater with the 
smaller sizes. 

The above applies to flat surfaces and refers to flat 
heads, such as dome heads, segments of heads, etc. 
The United States Government has a rule to find the 
pressure on flat heads not exceeding 20 in. in diameter 
as follows: 

CXT 2 
A 

where P = pressure. 

C=ii2 (7/16 or under) and 120 over 7/16. 

A = 1/2 the area. 

T = thickness in sixteenths. 
With a 3/4-in. head 20 in. in diameter, no lb. would 
be allowed by these rules. 

A short time ago the Board of Boiler Rules for the 
State of Ohio issued instructions to the inspectors holding 
certificates of competency that the following formula 
could be used in flat surface of heads: 

rxr.s.xio 

ax's 

w T here T = thickness. 

T.S. = tensile strength. 
A = area. 
In addition, a limitation allowance of 8000 lb. per 
square inch irrespective of size of brace is granted. 
This applies to boilers now in use, but not to boilers 
to be installed after July 1, 19 12. This ruling is far 
more liberal than any other authority has heretofore 



MISCELLANEOUS APPLICATIONS 123 

allowed as a comparison will show. Assume a 72-in. 
boiler, height of segment 24 in., thickness of head 1/2 
in., tensile strength 60,000 lb. The total area of the 
segment = 1 186.4 S Q- in., while the area requiring bracing 
= 814 sq. in. Hence 

.5X60000X10 r „ 

814x8 " =461b - 

allowed without braces. 

Let the pressure required be equal to 100, then 100 — 
46 = 54 lb. to be braced, and 54X814 = 43,956 lb. 
Assuming the proposed brace to be of .79 in. area, 
then .79X8000 = 6320 per brace, and 43,956^6320 = 7 
braces of practically i-in. diameter. 

It may be said that flat surfaces subjected to internal 
pressure will spring and proportionally to the unsup- 
ported area. Samuel Nichols, in his tests of circular 
flat heads, showed the springing began with very low 
pressures, even at 20 lb. on 28-in. heads and increased as 
the pressure was raised. Applying this fact then to the 
Ohio ruling, it seems the head would so spring that at 
100 lb. pressure the total load on the braces would be 
.79X7 = 5.53 into the toal load, 81,400 lb., or 14,718 
per square inch net section instead of 8000 lb. Applying 
this ruling to a flat dome head 36 in. diameter, 1/2 in. 
thick, 60,000 lb. tensile strength, area to be braced 
707 in., gives 53 lb. without bracing. The results of 
allowing a flat head unbraced 1 o spring and return times 
without number would be final failure due to such action. 

Reverting to the segment as in a horizontal tubular 
boiler, it may be said other authorities have been careful 



124 ARITHMETIC OF THE STEAM BOILER 

to avoid allowing excessive stresses on the chord of the 
segment which is supported by the tubes inasmuch as 
the latter are not a constant in strength as is the flange 
of the head in the arc of the segment, and this view has 
been approved by most students as the tubes are subjected 
to more or less rapid wear and reduction in thickness. 

Further, such calculations apply to boilers now in use 
irrespective of age. Indeed, Ohio has no limitation as 
to age as respects pressure to be determined with a factor 
of safety of 4 together with this exceedingly liberal allow- 
ance on braces. Comparing this with the Massachusetts, 
Chicago, Philadelphia and Detroit rules, w T hat results may 
be expected? 

Three Boiler Questions 

In an examination, three out of live engineers failed 
to answer the following questions, which are given for the 
benefit of those w T ho may be called upon to make similar 
calculations. 

1. A horizontal tubular boiler is 72 in. in diameter 
and 18 ft. long; thickness of plate .437 in.; efficiency of 
longitudinal joints 77 per cent., and steam pressure no 
lb. What should be the tensile strength of the plate, 
allowing a factor of safety of 5 ? 

2. If the tensile strength had been 56,000 lb. and 
the efficiency of the joint 70 per cent., what thickness of 
plate should be employed? 

3. If this boiler had been intended for 125 lb. of 
steam, what would the efficiency of the joint have been, 
using the data in the first question (except the pressure 
and efficiency) ? 



MISCELLANEOUS APPLICATIONS 125 

The required tensile strength of plate is found by the 
rule 

pressure X diam. X factor of safety 
efficiency of joint X thickness of plate X 2 

Substituting the figures given, instead of the words in 
the rule, we have 

Tensile strength = 1 -°~^ 2 —^ = 58,7 7 5 lb. 
.77X437X2 ° ,//0 

The rule for thickness of plate is 

pressure X diam. X factor of safety 

tensile strength X efficiency of joint X 2 

Again substituting the figures we have 

rrw 1 110X72X5 
inickness =—- — — — •= . c m. 

56000 X. 70X2 J 

Efficiency of joint is given by the rule 

. _ pressure X diam. X factor of safety 

^ tensile strength X thickness X 2 
This figured out gives 

12^ X 72 X ^ 

Efficiency = —z — — — = .87^ or 87. < per cent. 

58775X.437X2 /0 ' 0F 

To Find Pitch of Rivets 

How can the pitch of the rivets be determined for a 
double-riveted butt and double-strap joint which is to 
have 7/8-in. rivets and a strength of plate between the 
rivet holes on the outer row which will be 82 per cent, 
of the strength of the solid plate? 
Let P = pitch. 

t = thickness of plate. 
TS = tensile strength of plate. 
d = diameter of rivets. 



126 ARITHMETIC OF THE STEAM BOILER 

Then 

PXtXTS = strength of solid plate 
and 

(P — d) XtXTS = strength of plate between the rivet 
holes on the outer row. The conditions require 
(P-d)tXTS _ 
PXtXTS W 

By canceling out t and TS from numerator and de- 
nominator of the first member of (i) we obtain, 

P-d 



P 

and as ^ = .875, the equation becomes 

^-•875 
P 

from which it is found that 



= .82 (2) 

ecomes 

=•82 ( 3 ) 



P=--^= 4 .86+ (4) 

or practically 4 7/8 in. 

Collapsing Pressure of Lap-welded Bessemer 

Steel Tubes of from 3 to 10 In. Diameter, and 

of Different Wall Thicknesses 

Formulas: 



and 





t 






P-- 


= 86,670-7- 

/ 


-1386 






/ 


p\ 


p= 


- IOOO ( I — y 


J 1 " 1 


6oo j2 ) 


p= 


= 50,210,000 


(t)» 





MISCELLANEOUS APPLICATIONS 127 

where p = collapsing pressure in pounds per square inch. 
d = outside diameter of tube in inches. 
/ = wall thickness in inch measure. 

The first formula is applicable to cases where -j is 

greater than .023 and the others to the case of thin-walled 
tubes where the quotient is less than that value. 

Safe Working Pressure Calculations as Applied 

to the Shell of Climax, Hazelton and Porcupine 

Types of Steam Boilers 

Example. — Shell plate 5/8 in. thick, diameter 30 in., tensile 
strength 60,000 lb. per square inch, tubes 4 in. diameter. See Figs. 
1 and 2 of this section. 

Consider the ring of the shell 3 27/32 in. wide (Fig. 2) 
included between any two transverse rows of holes. 
For each pound per square inch of pressure, any longi- 
tudinal section of this ring will be subjected to a stress of 

= 57.65625 lb. The net section of the ring 

on the axis ab of a longitudinal row of holes is 5/8 X 
(3 27/32 — 2) = i. 15+sq. in. The unit stress on this 
section for a pressure of 1 lb. per square inch is, therefore, 
57.656-^1.15 = 50 lb. per square inch, nearly. The 
section on the line ac through two adjacent holes in a 
diagonal row is subjected to the stress of 57.656 lb., 
which acts in the direction of the line ef, perpendicular to 
ab. This stress may be resolved into two components, one 
of which, eg, acts perpendicular to ac and tends to pull the 
plate apart through that section, while the other, eh, acts 
along the line ac and tends to shear the plate through the 



128 



ARITHMETIC OF THE STEAM BOILER 



same section. Of these two components, eg is equal to 
the stress on a longitudinal section of the ring multiplied 
by the cosine of the angle bac, and eh is equal to the stress 
'on the longitudinal section multiplied by the sine of the 
angle bac. The sine of bac is 3.5-^5 • 2 = .673, and 
the cosine 3 27/32-7-5.2 = 
• 739? which correspond to an 
angle of 47 40', nearly. The 
tensile stress on the section 
ac resulting from the stress 
of 57.656 lb. acting perpen- 
dicular to the section ab is, 
therefore, 57.656 X .739 = 





Dia. of Shell 30" 
Thickness of Plate % 

Fig. i. — Calculations relating Fig. 2. — Calculations relating to 
to porcupine type of boilers. porcupine type of boilers. 

42.61 lb., nearly, and the shearing stress is 57. 656X^73 = 
38.8 lb. In addition to the stresses due to the force that 
tends to break the ring through a longitudinal section, 
the section on ac is subjected to a stress from the action 
of the force that tends to rupture the shell along a trans- 
verse section. For a pressure of 1 lb. per square inch, 



MISCELLANEOUS APPLICATIONS 129 

this force is equal to the area of the head multiplied by 1 ; 
that is, to 3o 2 X-7854X 1 = 706.86 lb. The number of 
sections among which this force is divided is equal to the 
circumference of a 30-in. circle divided by 3.5; that is, to 

^ — — =27. The force on each section is, there- 

3-5 
fore, 706.86-^27 = 26.18 lb. This force acts on the 

section ac in the direction of the line el, parallel to the 
line ab. It may be resolved into two components, one, 
em, perpendicular to ac, which is equal to 26.18 multi- 
plied by the sine of the angle bac; the other, en, in the 
direction of ac, equal to 26.18 multiplied by the cosine of 
bac. Of these two components, the first acts in the same 
direction as the component eg; its value, 26.i8X.673 = 
17.62 lb., nearly, is, therefore, to be added to the value 
represented by eg, thus giving us a total tensile stress in 
the section ac of 42.61 + 17.62 = 60.23 lb. The compo- 
nent en, whose value is 26. 18X. 739 = 19. 347 lb., acts in 
the opposite direction to eh; they therefore partly neu- 
tralize each other, and the resulting shearing stress, 
38.8 — 19.347 = 18.453, is so much less than the tensile 
stress of 60.23 lb. that it is evident the section would 
fail by tension and not by shear. The area of the net 
section of the plate, which resists the tensile stress of 
60.23 lb., is 1.2X5/8 = 75 sq. in.; the unit stress in this 
section for a pressure of 1 lb. per square inch is, therefore, 
60.23-^.75 = 80.3 lb. per square inch. Since the stress on 
the section ab was but 50 lb. per square inch, it is evident 
that the plate will fail along the section ac. If we 
assume the safe working stress of the 60,000-lb. steel 
plate to be 10,000 lb. per square inch, the safe working 



130 



ARITHMETIC OF THE STEAM BOILER 



pressure will be 10,000^80.3 = 124.5 lb. per square 
inch. 

Figuring the Safe Working Pressure op the Shell 
of a Locomotive Boiler 

How to determine the safe working pressure of the 
shell of a locomotive boiler with several courses of vary- 
ing diameter, like that shown in Fig. 21. 

With the locomotive boiler, like Fig. 21, the safe work- 
ing pressure can only be ascertained by considering the 




Fig. 21. — Diagram from which to calculate the safe pressure in a 
locomotive boiler. 



diameters A, B and C. Also, the thickness of the plates 
and the efficiencies of the longitudinal seams of the re- 
spective courses must be considered. 
! The inside diameter A is 72 in. and plate 3/4 in. in 
thickness, the inside diameter B 61 in. and plate 9/16 
in. in thickness, and the inside diameter C 60 in. and 
plate 1/2 in. in thickness. The efficiency of the riveted 
joints for the respective courses is A, 82 per cent., course 
B 84 per cent., and course C 86 per cent. 

It may be asked, what is the difference in efficiency in 



MISCELLANEOUS APPLICATIONS 131 

the respective courses? This is because the over-all 
distances, D, E and F, are such that the same maximum 
pitch could not be obtained in the respective courses, 
and a change of pitch, large or small, will make a dif- 
ference in the efficiency of the net section of plate, maxi- 
mum pitch of rivets, which point is made the weaker of 
the several parts of a riveted joint. 

Assuming the factor of safety to be 5, and the plate 
to ha\e a tensile strength of 60,000 lb. then the working 
pressure of the boiler, as far as the shell is concerned, 
may be determined by the following formula: 
Where T — thickness of the plate in inches. 

D = diameter of the boiler in inches. 

T s = tensile strength of the plate in pounds. 

F = factor of safety. 

E = efficiency of the longitudinal seam. 

P = pressure in pounds per square inch. 

TXT S X E_ 
DXF 

The safe working pressure for course A then will be 

(2X3/4)X6ooooX.82 

— — = 205 lb. 

72X5 D 

The safe working pressure for course B then will be 

(2X9/i6)X6ooooX.8 4 

y-^~ = 186 lb. 

61X5 

The safe working pressure for course C then would be 

(2Xi/2)X6ooooX.86 

, ., — = 172 lb. 

60X5 ' 



132 ARITHMETIC OF THE STEAM BOILER 

The calculations thus show that the course C, the 
course with the least diameter and the longitudinal seam 
with the greatest efficiency, to be the weaker of the three 
courses A, B and C. Therefore, the pressure for the 
boiler, considering the shell only, would be 172 lb. 

Had the designer in the first instance made the course 
C 9/16 plate and the course B 5/8-in. plate, the boiler 
shell in question could have been allowed a greater 
pressure than 172 lb. To a boiler designer these calcu- 
lations would suggest several things. First would be 
that if no change was to be made in course C, then the 
thickness of course A could be reduced, perhaps, 1/16 
in. in thickness. This would make a saving in the cost 
of the boiler. Second, if need be the efficiencies of the 
longitudinal seams of courses A and B could be less. 
This is, of course, on the assumption that course A will 
undergo no changes in regard to thickness of plate, di- 
ameter, efficiency of longitudinal seam and tensile 
strength of plate. 



PART III 

APPENDIX 

EXTRACTS FROM UNITED STATES RULES- 
MARINE— AND FROM THE BOARD OF 
BOILER RULES STATE OF MASSACHU- 
SETTS—TABLES 



APPENDIX 

EXTRACTS FROM RULES OF THE UNITED 
STATES BOARD OF SUPERVISING INSPEC- 
TORS, STEAMBOAT INSPECTION SERVICE 

United States Rules Pertaining to Riveted Joints 

The following formulas, equivalent to those of the 
British Board of Trade, are given for the determination 
of the pitch, distance between rows of rivets, diagonal 
pitch, maximum pitch, and distance from centers of 
rivets to edge of lap of single- and double-riveted lap 
joints, for both iron and steel boilers. 

Let p = greatest pitch of rivets in inches. 
n = number of rivets in one pitch. 
pd = diagonal pitch in inches. 
d = diameter of rivets in inches. 
T — thickness of plate in inches. 
V = distance between rows of rivets in inches. 
E = distance from edge of plate to center of rivet 
in inches. 

TO DETERMINE THE PITCH 

Iron plates and iron rivets: 

<PX. 78 54X7* , . 
p=~ —jr- - +d 

135 



136 ARITHMETIC OF THE STEAM BOILER 

Example, first, for single-riveted joint: Given, thickness of plate 
(r) = i/2 in., diameter of rivet (d) = 7/8 in. In this case n = i. 
Required the pitch. 

Substituting in formula, and performing operation indicated, 

p . + . (7/8) 2 X. 7854X1 , ._ 

Pitch = j + 7/8 =2.077 in. 

Example for double-riveted joint: Given, / = 1/2 in., andd = 13/16 
in. In this case, n = 2. Then — 

pitch = (MA6)!>C7854>0 ^ 

1/2 

For steel plates and steel rivets: 

23Xd 2 X. 7854X7* . , 



P 



28XT 



Example for single-riveted joint: Given, thickness of plate = 
1/2 in., diameter of rivet 15/16. In this case, n — \. 

p ., , 2 3 Xd5/i6) 2 X. 7854X1 . 

Pitch = 28X1/2 1-15/16 =2.071 in. 

Example for double-riveted joint: Given, thickness of plate = 1/2 
in., diameter of rivet = 7/8 in. 71 — 2. Then — 

p-*i. 2 3X(7/8) 2 X. 7854X2 

Pitch = 28Xl/2 "+7/8 =2.85 in. 



FOR DISTANCE FROM CENTER OF RIVET TO EDGE OF LAP 

E= sXd 

2 

Example. — Given, diameter of rivet (d) = 7/8 in., required the 
distance from center of rivet to edge of plate. 

E= — =1.312 in., for single- or double-riveted lap joint. 



APPENDIX 137 



FOR DISTANCE BETWEEN ROWS OF RIVETS 

The distance between lines of centers of rows of rivets 
for double, chain-riveted joints (V) should not be less 
than twice the diameter of rivet, but it is more desirable 

4J+1 
that V should not be less than * 

2 

Example under latter formula: Given, diameter of rivet = 7/8 in.; 
then — 

( 4 X7/8) + i 
V= =2.25 in. 

2 ° 

For ordinary, double, zigzag riveted joints: 

v _\/ (iip+4d)(p+4d) 
10 

Example. — Given, pitch = 2.85 in. and diameter of rivet = 7/8 
in.; then — * 

T/-V / (nX2.85+4X7/8) (2.85+4X7/8) . . 

V— =1.487 m. 



DIAGONAL PITCH 

For double, zigz&g riveted lap joint. Iron and steel: 

. 6p+ 4 d 

pd= 

10 

Example. — Given, pitch = 2.85 in., and d = j/& in.; then — 

A (6X2.85)+ (4X7/8) 

pd= = 2.00 in. 

r 10 

1 Extract the square root of the expression above the line only, 
then divide by 10. 



138 ARITHMETIC OF THE STEAM BOILER 

Maximum Pitches for Riveted Lap Joints 
For single-riveted lap joints: 

Maximum pitch =(1.31 XT) + 1 s/8.J 
For double-riveted lap joints: 

Maximum pitch = (2.62 XT) + 1 5/8. 

Example. — Given, a thickness of plate = 1/2 in., required the maxi- 
mum pitch alio: -able. 

For single-riveted lap joint: 

Maximum pitch = (1.31X1/2)4-1 5/8 = 2.28 in. 

For double-riveted lap joint: 

Maximum pitch= (2.62X1/2) + ! 5/8 = 2.935 in. 

TO DETERMINE THE AREAS OF DIAGONAL STAYS 

Multiply the area of a direct stay required to support 
the surface by the slant or diagonal length of the stay; 
divide this product by the length of a line drawn at right 
angles to surface supported to center of palm of diagonal 
stay. The quotient will be the required area of the 
diagonal stay. 

aXL 



A = 



I 



Where A = sectional area of diagonal stay. 
fa = sectional area of direct stay. 
L — length of diagonal stay. 
1 = length of line drawn at right angles to boiler 
head or surface supported to center of palm 
of diagonal stay. 



APPENDIX 139 

Given diameter of direct stay = i in., a = .7845, L = 6o 
in., 1 = 48 in., substituting and solving, 

.7854X60 

^4 = q — - = .981 sectional area. 

45 

Diameter — 1 . 1 1 in. = 1 1/8 in. 

The sectional area of gusset stays, when constructed 
of triangular right-angled web plates secured to single 
or double angle bars along the two sides at right angles, 
shall be determined by formula for diagonal stays, and 
shall not be less than 10 per cent, greater than would 
be necessary for a diagonal bolt stay. 



STAYS 

The maximum stress in pounds allowable per square 
inch of cross-sectional area for stays used in the con- 
struction of marine boilers, when same are accurately 
fitted and properly secured, shall be ascertained by the 
following formula: 

p= AXC 
a 

Where P = working pressure in pounds. 

A= least cross-sectional area of stay in inches. 

a = area of surface supported by one stay in 
inches. 

C = a constant. 

C = 9ooo for tested steel stays 1 1/4 in. and 
upward in diameter, when such stays are not 
forged or welded. The ends may be upset to a 



140 ARITHMETIC OF THE STEAM BOILER 

sufficient diameter to allow for the depth of 
the thread, provided it is the least diameter of 
the stay. All such stays after being upset 
shall be thoroughly annealed. 

C = 8ooo for a tested Huston or similar type of 
brace, the cross-sectional area of which ex- 
ceeds 5 sq. in. 

C = 7ooo for such tested braces when the cross- 
sectional area is not less than 1.227 and not 
more than 5 sq. in., provided such braces 
are prepared at one heat from a solid piece 
of plate without welds. 

C = 75oo for wrought iron through stays 1 1/4 
in. diameter and upward. When made of 
the best quality of refined iron, they may be 
welded. 

C = 6ooo for welded crowfoot stays when made 
of the best quality of refined wrought iron, 
and for all stays not otherwise provided for 
when made of the best quality of refined 
iron or steel without welds. 

Furnaces 

The tensile strength of steel used in the construction 
of corrugated or ribbed furnaces shall not exceed 67,000 
and be not less than 54,000 lb.; and in all other furnaces 
the minimum tensile strength shall not be less than 
58,000 and the maximum not more than 67,000 lb. 
The minimum elongation in 8 in. shall be 20 per cent. 

All corrugated furnaces having plain parts at the ends 



APPENDIX 141 

not exceeding 9 in. in length (except flues especially pro- 
vided for) when new, and made to practically true circles, 
shall be allowed a steam pressure in accordance with the 
following formula: 

CXT 



P = 



D 



LEEDS SUSPENSION BULB FURNACE 

CXT 



P = 



D 



Where P = pressure in pounds. 

T = thickness in inches, not less than 5/16 in. 

Z) = mean diameter in inches. 

C = a constant, 17,300, determined from an actual 
destructive test under the supervision of the 
Board, when corrugations are not more than 
8 in. from center to center, and not less than 
2 1/4 in. deep. 

MORISON CORRUGATED TYPE 

CXT 



p = 



D 



Where P = pressure in pounds. 

T = thickness in inches, not less than 5/16 in. 

Z) = mean diameter in inches. 

C = 15,600, a constant, determined from an 
actual destructive test under the supervision 
of the Board of Supervising Inspectors, 



142 ARITHMETIC OF THE STEAM BOILER 

when corrugations are not more than 8 

in. from center to center and the radius 

of the outer corrugations is not more 

than one-half of the suspension curve. 

[In calculating the mean diameter of the Morison 

furnace, the least inside diameter plus 2 in. may be taken 

as the mean diameter, thus — 

Mean diameter = least inside diameter +2 in.] 

FOX TYPE 

CXT 



F = 



D 



Where P = pressure in pounds. 

T = thickness in inches, not less than 5/16 in. 

Z) = mean diameter in inches. 

C — 14,000, a constant, when corrugations are 
not more than 8 in. from center to center 
and not less than 1 1/2 in. deep. 

PURVES TYPE 

CXT 



P = 



D 



Where P = pressure in pounds. 

T = thickness in inches, not less than 7/16 in. 
Where D== least outside diameter in inches. 

C= 14,000, a constant, when rib projections are 
not more than 9 in. from center to center 
and not less than 1 3/8 in. deep. 



APPENDIX 143 



BROWN TYPE 

p= CXT 
D 

Where P = pressure in pounds. 

T = thickness in inches, not less than 5/16. 
D = least outside diameter in inches. 
C= 14,000, a constant (ascertained by an 
actual destruction test under the supervision 
of this Board), when corrugations are not 
more than 9 in. from center to center and 
not less than 1 5/8 in. deep. 
The thickness of corrugated and ribbed furnaces shall 
be ascertained by actual measurement. The manufac- 
turer shall have said furnace drilled for a 1/4-in. pipe 
tap and fitted with a screw plug that can be removed 
by the inspector when taking this measurement. For the 
Brown and Purves furnaces the holes shall be in the 
center of the second flat; for the M orison, Fox, and other 
similar types in the center of the top corrugation, at 
least as far in as the fourth corrugation from the end of 
the furnace. 

TYPE HAVING SECTIONS 1 8 IN. LONG 

p _CXT 
D 

Where P = pressure in pounds. 

T — thickness in inches, not less than 7/16. 
D = mean diameter in inches. 



144 ARITHMETIC OF THE STEAM BOILER 

C= 10,000, a constant, when corrugated by sec- 
tions not more than 18 in. from center to 
center and not less than 21/2 in. deep, measur- 
ing from the least inside to the greatest 
outside diameter of the corrugations, and hav- 
ing the ends fitted one into the other and 
substantially riveted together, provided that 
the plain parts at the ends do not exceed 12 
in. in length. 

TOPS OF COMBUSTION CHAMBERS AND BACK CONNECTIONS 

Formula for girders over back connection and other 
flat surfaces: 

Working pressure = (w _ F) ^p^Z 

Where W = width of combustion box in inches. 
P = pitch of supporting bolts in inches. 
D = distance between girders from center to cen- 
ter in inches. 
L = length of girder in feet. 
d = depth of girder in inches. 
T = thickness of girder in inches . 
C = 5So when the girder is fitted with one sup- 
porting bolt. 
C = 825 when the girder is fitted with two or 

three supporting bolts. 
C = 9i7 when the girder is fitted with four or five 

supporting bolts. 
C = 963 when six or seven supporting bolts are 

used. 



APPENDIX 145 

C = ggo when eight or more supporting bolts are 
used. 

EXAMPLE 

Given W = S4 in., -P=7-5 i n -> D=y.7S in., L = 2.927 ft., ^=7.5 
in., T=2 in., C = 82$, then, substituting in formula, 

w , • 825X7-5X7-5X2 _ 

\\ orkmg pressure = 7 ;— — 154-3 lb. 

F (34-7-5)X7-75X2. 9 27 

FLAT SURFACES 

The maximum stress allowable on flat plates sup- 
ported by stays shall be determined by the following 
formula : 

All stayed surfaces formed to a curve the radius of 
which is over 21 in., excepting surfaces otherwise pro- 
vided for, shall be deemed flat surfaces. 

W T CXr2 

Working pressure = — p 2 — 

Where T = thickness of plates in sixteenths of an inch. 
P = greatest pitch of stays in inches. 
C=ii2 for screw stays with riveted heads, 

plates 7/16 in. thick and under. 
C=i2o for screw stays with riveted heads, 

plates above 7/16 in. thick. 
C=i2o for screw stays with nuts, plates 7/16 

in. thick and under. 
C=i25 for screw stays with nuts, plates above 

7/16 in. thick and under 9/16 in. 
C=i35 for screw stays with nuts, plates 9/16 
. in. thick and above. 



146 ARITHMETIC OF THE STEAM BOILER 

C=i75 for stays with double nuts having one 
nut on the inside and one nut on the out- 
side of plate, without washers or doubling 
plates. 

C= 1 60 for stays fitted with washers or doubling 
strips which have a thickness of at least .5 
of the thickness of the plate and a diameter 
of at least .5 of the greatest pitch of the 
stay, riveted to the outside of the plates, 
and stays having one nut inside of the plate, 
and one nut outside of the washer or 
doubling strip. For T take 72 per cent, of 
the combined thickness of the plate and 
washer or plate and doubling strip. 

C = 200 for stays fitted with doubling plates 
which have a thickness equal to at least 
.5 of the thickness of the plate reenforced, 
and covering the full area braced (up to the 
curvature of the flange if any) riveted to 
either the inside or outside of the plate, 
and stays having one nut outside and one 
inside of the plates. Washers or doubling 
plates to be substantially riveted. For T 
take 72 per cent, of the combined thickness 
of the two plates. 

C=2oo for stays with plates stiffened with tees 
or angle bars having a thickness of at 
least two-thirds the thickness of plate and 
depth of webs at least one-fourth of the 
greatest pitch of the stays, and substan- 
tially riveted on the inside of the plates, 



APPENDIX 147 

and stays having one nut inside bearing 
on washers fitted to the edges of the webs 
that are at right angles to the plate. For 
T take 72 per cent, of the combined thickness 
of web and plate. 
No such flat plates or surfaces shall be unsupported at 
a greater distance than 18 in. 

REQUIREMENTS FOR HEADS 

All plates used as heads, when new and made to 
practically true circles, and as described below, shall be 
allowed a steam pressure in accordance with the following 
formula: 

CONVEX HEADS 

TXS 



P = 



R 



Where P = steam pressure allowable in pounds. 
T = thickness of plate in inches. 
5 = one-fifth of the tensile strength. 
R = one-half of the radius to which the head is 
bumped. 

CONCAVE HEADS 

For concave heads the pressure allowable will be .8 
times the pressure allowable for convex heads. 

Note. — To find the radius of a sphere cf which the bumped head 
forms a part, square the radius of head, divide this by the height of 
bump required; to the result add height of bump, which will equal 
diameter of sphere, one-half of which will be the required radius, 
n 



148 ARITHMETIC OF THE STEAM BOILER 

Example. — Required the working pressure of a convex head of a 

54-in. radius, material 60,000 lb. tensile strength and 1/2 in. thick. 

Substituting values, 

d -5X12000 

P = = 222 lb. 

27 

The pressure allowable on a concave head of the same dimensions 
would be 

222 X. 8 = 177 lb. 

ANGLE STIFFENERS FOR CURVED SURFACES 

Where rounded bottoms of combustion chambers are 
stiffened with single angle-iron stiffeners, such angles 
shall have a thickness of leaf eight-tenths that of the 
plate and a depth of at least one-half pitch. Where 
stiffened with double angle irons or tee bars, such angles 
or tee bars shall have a thickness of leaf at least two- 
thirds that of plate and a depth of at least one-fourth of 
pitch. Said angles or tee bars shall be substantially 
riveted to the plate supported. 

Where rounded tops of combustion chambers are 
stiffened with single or double angle-iron stiffeners, or 
tee bars, such angles or tee bars shall be of thickness and 
depth of leaf not less than specified for rounded bottoms 
of combustion chambers. Said angles or tee bars shall 
be supported on thimbles and riveted through with 
rivets not less than 1 in. in diameter, and spaced not 
to exceed 6 in. between centers. 

Working pressure allowed on rounded surfaces sup- 
ported by angle irons or tee bars shall be determined by 
the following formula: 

W T CXT2 

Working pressure = p jz 



APPENDIX 149 

Where T = thickness of plate in sixteenths of an inch. 
P = pitch of angle or tee stiffeners in inches. 
D = diameter of curve to which plate is bent, in 

inches. 
C = 9oo, a constant. 

Example. — Given T = g/i6 in. P = 7 in. D = s i in- 
Substituting values in formula and solving, 

^ 1 • 900X81 . 

\\ orkmg pressure = — — =20410. per square men 

PRESSURE PERMISSIBLE ON ROUNDED BOTTOM OF COM- 
BUSTION CHAMBERS, ANGLES BEING OMITTED 

5o(3oor- 2Z, ) 
' D 

Where P = working pressure in pounds. 

T= thickness of bottom plate of combustion 

chamber in inches. 
L = extreme length of plate forming bottom of 

combustion chamber in inches. 
D = twice outside radius of bottom of combustion 
chamber in inches. 

Example. — Required the working pressure on the bottom 
plate of a combustion chamber, angles being omitted: Thickness 
of plate, .82 in., extreme length of plate, 33 in., twice the radius 
of bottom of combustion chamber, 50 in. Substituting: 

p _ 5oX(3QoX.82-2X33) = l8o i b . 
50 
T _ PXD+icoL 

15000 



ISC ARITHMETIC OF THE STEAM BOILER 

Pressure allowable on tube sheets where combustion 
chambers are not suspended from the shell of the boiler 
shall be determined by the following formula: 

(D-d)XTX2 7 ooo 
WXD 

Where P = working pressure in pounds. 

D = least horizontal distance between tube centers, 
in inches. 
d = inside diameter of tubes in inches. 
T = thickness of tube plates in inches. 
W = extreme width of combustion chamber in 
inches. 

The compressive stress on tube plates, as determined 
by the following formula, must not exceed 13,500 lb. 
per square inch, when pressure on top of combustion 
chamber is supported by vertical plates of such 
chamber. 

PXDXW 

L 2(D-d)T 

Where C = stress on tube sheet. 

P = working pressure in pounds. 
D = least horizontal distance between tube cen- 
ters in inches. 
d = inside diameter of tubes in inches. 
W = extreme width of combustion chamber in 

inches. 
T = thickness of tube sheet in inches. 



APPENDIX 151 

Safety Valves 

The areas of safety valves shall be determined in 
accordance with the following formula and table: 

W 

# = .2074Xp- 

Where a = area of safety valve, in square inches, per 
square foot of grate surface. 
PF = pounds of water evaporated per square foot 

of grate surface per hour. 
P = absolute pressure per square inch = working 
gage pressure+15. 
From which formula the areas required per square 
foot of grate surface in the following table are found by 
assuming the different values of W and P. 

The figures (a) in table multiplied by square feet of 
grate surface give the area of safety valve or valves 
required. 

When this calculation results in an odd size of safety 
valve, use next larger standard size. 

Examples. — Boiler pressure = 75 lb. per square inch (gage). 

2 furnaces: Grate surface = 2 (No.)X5 ft. 6 in. (long)X3 ft. 
(wide) =33 sq. ft. 

Water evaporated per pound of coal = 8 lb. 

Coal burned per square foot grate surface per hour =12 1/2 lb. 

Evaporation per square foot grate surface per hour = 8X12 1/2 
= 100 lb. 

Hence W = 100 and gage pressure = 75 lb. 

From table the corresponding value of a is .230 sq. in. 

Therefore area of safety valve = 33 X. 23 = 7.59 sq. in. 

For which the diameter is 3 1/8 in. nearly. 

Boiler pressure = 215 lb. 



152 



ARITHMETIC OF THE STEAM BOILER 



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APPENDIX 



153 






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154 ARITHMETIC OF THE STEAM BOILER 

6 furnaces: Grate surface = 6 (No.)Xs ft. 6 in. (long) X 3 ft. 4 in. 
(wide) = 110 sq. ft. 

Water evaporated per pound coal = 10 lb. 

Coal burned per square foot grate surface per hour = 30 lb. 

Evaporation per square foot grate surface per hour = 10X30 = 
300 lb. 

Hence IF = 300, gage pressure = 215, and a = .2 70 (from table). 

Therefore area of safety valve = noX.2 7o = 29.7 sq. in., which 
is too large for one valve. Use two. 

20. 7 

= 14.85 sq. in. Diameter = 4 3/8 in. 

To determine the area cf a safety valve for boiler using oil as 
fuel or for boilers designed for any evaporation per hour, 

Divide the total number of pounds of water evaporated per hour 
by any number of pounds of water evaporated per square foot of 
grate surface per hour (W) taken from, and within the limits of, 
the table. This will give the equivalent number of square feet of 
grate surface for boiler; for estimating the area of valve, then apply 
the table as in previous examples. 

Example. — Required the area of a safety valve for a boiler using 
oil as fuel, designed to evaporate 8000 lb. of water per hour, at 
175-lb. gage pressure. 

Make W = 200. 

8000 . 

— =40, the equivalent 
200 ^ ' n 

grate surface in square feet. 

For gage pressure = 175 lb. and W = 2oo, from table, a = .218 sq. 
in.; .218X40 = 8.72 sq. in., the total area required for this boiler, 
for which the diameter is 3 5/16 in. closely. 



WATER TUBE AND COIL BOILERS 

The working pressure allowable on cylindrical shells of 
water tube or coil boilers, when such shells have a row 



APPENDIX 155 

or rows of pipes or tubes inserted therein, shall be deter- 
mined by the following formula: 

(D-d )XTXS 
DXR 

Where P = working pressure allowable in pounds. 

D = distance in inches between the tube or pipe 
centers in a line from head to head. 

d = diameter of hole in inches. 

T = thickness of plate in inches. 

S = one-sixth of the tensile strength of the plate. 

R = radius of shell in inches. 

n = number of tube holes in a pitch. When tubes 
on any one row are pitched unequally, nd 
must be substituted in the formula for d; 
where rows of tubes are pitched diagonally, 
each diagonal ligament shall not be less than 
three-fifths of each longitudinal ligament. 

Example. — Required the working pressure of a cylindrical shell 
having holes i in. in diameter, spaced 2 in. from center to center, in 
a line from head to head; material, 1/2 in. thick; diameter of shell, 
20 in.; tensile strength of plate, 60,000 lb. 

Substituting values, we have 

(2 -i)X. 5X10000 
P=— — — = 250 lb. 

2X10 ° 

PORCUPINE-TYPE BOILERS 

The formula for determining pressure on boilers of the 
so-called Porcupine and similar types shall be as follows: 

Multiply the vertical distance between the centers of 
the horizontal rows of tubes in inches by one-half the 
diameter of shell of boiler in inches, which gives the area 



156 ARITHMETIC OF THE STEAM BOILER 

upon which the pressure is exerted to break a diagonal 
ligament, then find the sectional area of the ligament at 
its smallest part and multiply by one-sixth the tensile 
strength of the material. This result, divided by the 
area upon which the strain is exerted, gives the working 

EFT 

pressure per square inch, which is as follows =W, 

the working pressure, in which E equals wddth of liga- 
ment in inches, F thickness of material in inches, T one- 
sixth of the tensile strength, C distance between vertical 
centers, and D one-half the inside diameter of the shell 
or central column. 

For the boiler proposed, 30 in. diameter, 5/8 in. 
thick, tensile strength 60,000 lb., 1.219 in. would be 
width of ligament, .625 thickness of plate, 10,000 
one-sixth of tensile strength, 3 11/16 = 3.6875 in., 
distance of vertical centers; 15 in., one-half the diam- 
eter of shell, would be as follows: 1.219 multiplied by 
.625, this product multiplied by one-sixth the tensile 
strength, 10,000, equals 7618.75. This product, divided 
by the product of 3.6875, distance between vertical 
centers, multiplied by 15, one-half the diameter, equals 
55.3125, gives 137.7 as pressure allowed. 

EXTRACTS FROM BOARD OF BOILER RULES, 
STATE OF MASSACHUSETTS 

Maximum Pressure on Boilers 

1. The maximum pressure allowed on any steam boiler 
constructed wholly of cast-iron shall not be greater 
than twenty-five (25) pounds to the square inch. 



APPENDIX 157 

2. The maximum pressures allowed on any steam 
boiler, the tubes of which are secured to cast-iron headers, 
shall not be greater than one hundred and sixty (160) 
pounds to the square inch. 

3. The maximum pressure allowed on any steam boiler 
constructed of iron or steel shells or drums shall be 
calculated from the inside diameter of the outside course, 
the percentage of strength of the longitudinal joint and 
the minimum thickness of the shell plates; the tensile 
strength of shell plates to be taken as fifty-five thousand 
(55,000) pounds per square inch for steel and forty-five 
thousand (45,000) pounds per square inch for iron 
when the tensile strength is not known. 

SHEARING STRENGTH OF RIVETS 

4. The maximum shearing strength of rivets per square 
inch of cross- section of area to be taken as follows: 

Pounds 

Iron rivets in single shear 38,000 

Iron rivets in double shear 70,000 

Steel rivets in single shear 42,000 

Steel rivets in double shear 78,000 

Factors of Safety 

5. The lowest factors of safety used for steam boilers, 
the shells or drums of which are directly exposed to the 
products of combustion and the longitudinal joints of 
which are of lap-riveted construction, shall be as follows: 

(a) Five (5) for boilers not over ten years old. 

(b) Five and five-tenths (5.5) for boilers over ten and 
not over fifteen years old. 






158 ARITHMETIC OF THE STEAM BOILER 

(c) Five and seventy-five hundredths (5.75) for boilers 
over fifteen and not over twenty years old. 

(d) Six (6) for boilers over twenty years old. 

(e) Five (5) on steam boilers, the longitudinal joints 
of which are of lap-riveted construction, and the shells 
of drums of which are not directly exposed to the product s 
of combustion. 

(f) Four and five-tenths (4.5) on steam boilers, the 
longitudinal joints of which are of butt and strap 
construction. 

Fusible Plugs 

1. Fusible plugs as required by Section 20, Chapter 
465, Acts of 1907, shall be filled with pure tin. 

2. The least diameter of fusible metal shall not be less 
than one-half inch, except for working pressure of over 
one hundred and seventy-five (175) pounds gage, or 
when it is necessary to place a fusible plug in a tube, in 
which cases the least diameter of fusible metal shall not 
be less than three-eighth (3/8) inch. 

3. The location of fusible plugs shall be as follows: 

(a) In Horizontal Return Tubular Boilers — in the back 
head, not less than two (2) inches above the upper row of 
tubes, and projecting through the sheet not less than one 
(1) inch. 

(b) In Horizontal Flue Boilers — in the back head, on 
a line with the highest part of the boiler exposed to the 
production of combustion, and projecting through the 
sheet not less than one (1) inch. 

(c) In Locomotive Type or Star Water Tube Boilers — 



APPENDIX 159 

in the highest part of the crown sheet, and projecting 
through the sheet not less than one (1) inch. 

(d) In Vertical Fire Tube Boilers — in an outside tube, 
placed not less than one-third (1/3) the length of the 
tube above the lower tube sheet. 

(e) In Vertical Submerged Tube Boilers — in the upper 
tube sheet. 

(f) In Water Tube Boilers, Horizontal Drums, 
Babcock & Wilcox Type — in the upper drum, not less than 
six (6) inches above the bottom of the drum, and over the 
first pass of the products of combustion, projecting 
through the sheet not less than one (1) inch. 

(g) In Stirling Boilers, Standard Type — in the front 
side of the middle drum, not less than six (6) inches 
above the bottom of the drum, and projecting through 
the sheet not less than one (1) inch. 

(h) In Stirling Boilers, Superheater Type — in the 
front drum, not less than six (6) inches abo\e the bottom 
of the drum, and exposed to the products of combus- 
tion, projecting through the sheet not less than one 
(1) inch. 

(i) In Water Tube Boilers, Heine Type — in the front 
course of the drum, not less than six (6) inches above 
the bottom of the drum, and projecting through the 
sheet not less than one (1) inch. 

(j) In Robb-Mumford Boilers, Standard Type — in 
the bottom of the steam and water drum, twenty-four 
(24) inches from the center of the rear neck, and pro- 
jecting through the sheet not less than one (1) inch. 

(k) In Water Tube Boilers, Almy Type — in the tube 
directly exposed to the products of combustion. 



160 ARITHMETIC OF THE STEAM BOILER 

(1) In Vertical Boilers, Climax or Hazelton Type — 
in a tube or center drum not less than one-half (1/2) 
the height of the shell, measuring from the lowest cir- 
cumferential seam. 

(m) In Cahall Vertical Water Tube Boilers — in the 
inner sheet of the top drum, not less than six (6) inches 
above the upper tube sheet. 

(n) In Scotch Marine Type Boilers — in combustion 
chamber top, and projecting through the sheet not less 
than one (1) inch. 

(o) In Dry Back Scotch Type Boilers — in rear head, 
not less than two (2) inches above the top row of tubes, 
and projecting through the sheet not less than one (1) 
inch. 

(p) In Economic Type Boilers — in the rear head above 
the upper row of tubes. 

(q) In Cast-iron Sectional Heating Boilers — in a 
section over and in direct contact with the products of 
combustion in the primary combustion chamber. 

(r) For other types and new designs, fusible plugs 
shall be placed at the lowest permissible water level, in 
the direct path of the products of combustion, as near 
the primary combustion chamber as possible. 



Size of Rivets 

1. When the size of the rivets in the longitudinal 
joints of a boiler is not known, the diameter and cross- 
sectional area of rivet, after driving, shall be taken as 
follows : 



APPENDIX 



161 



Thickness of plate. I7/16 in.i7/i6 in.l 15/32 in.l 1/2 in. |q/i6 in. | 5/8 in. 





7/8 in. 


15/16 


15/16 in. 


15/16 


1 1/16 


I 1/16 


Diameter of rivet 


up to 


in. over 




in. 


in. 


in. 


after driving. 


2 1/4 in. 
pitch 


2 1/4 in. 
pitch 










Cross-sectional area of 


.6013 


.6903 


.6903 


.6903 


.8866 


.8866 


rivet after driving. 


sq. in. 


sq. m. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 



Thickness of plate. 


1/4 
in. 


9/32 
in. 


5/i6 
in. 


11/32 
in. 


3/8 
in. 


3/8 
in. 


13/32 
in. 




1 1/16 


1 1/16 


3/4 


3/4 


3/4 in. 


13/16 in. 


13/16 


Diameter of rivet 


in. 


in. 


in. 


in. 


up to 


over 


in. 


after driving. 










2 in. 
pitch 


2 in. 
pitch 




Cross-sectional 


• 3712 


.3712 


.4418 


.4418 


.4418 


.5185 


.5185 


area of rivet after 


sq. in. 


sq. in. 


sq. m. 


sq. m. 


sq. 111. 


sq. in. 


sq. in. 


driving. 

















Allowable Strain on Stays 

1. The maximum allowable strain per square inch net 
cross-section for weldless mild steel shall be as follows: 



Size up to and in- 
Type eluding 1 1/2 in. diam- 
eter or equivalent 


Size over 1 1/2 in. 
diameter or equiva- 
lent 


Head to head or through stays. 


8,000 lb. 


9,000 lb. 


Diagonal or crowfoot stays. . . . 


7,500 lb. 


8,000 lb. 


Screwed stays (stay bolts) 


7,000 lb. 


7,000 lb. 



2. For welded stays the strain allowed per square inch 
net cross-section shall not exceed six thousand (6000) 
pounds. 

3. For wrought-iron stays or stay bolts the strain 
allowed per square inch net cross-section shall not 
exceed six thousand (6000) pounds. 



l62 



ARITHMETIC OF THE STEAM BOILER 



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APPENDIX 



163 



Appendages to be Placed on Boilers 

1. Each boiler shall have a safety valve the minimum 
area of which shall be in accordance with the following 
tables. If more than one safety valve is used the mini- 
mum combined area shall be in accordance with the 
following tables. 

2. When the conditions exceed those on which the tables 
are based the formula shall be used. 

3. A table of areas of grate surface in square feet for 
pop safety valves follows: 



A = Area of safety valve in square inches per square 

foot of grate. 
W = Weight of steam per second. 
P = Pressure, absolute. 
4. A table of grate areas in square feet for safety 
valves (other than pop safety valves) follows; this table 
is in ratio to the table for pop safety valves as 2 is to 3 : 



Gage pressure per square inch at 


Zero to 


Over 25 to 


Over 50 to 


which safety valve is set to blow 


25 lb. 


50 lb. 


100 lb. 


Diameter of valve 
in inches 


Area of valve in 
square inches 


Area of grate in sqt 


iare feet 


1 


• 7854 


1.4 


1.6 


1.8 


1 1/4 


1 .2272 


2 . 1 


2.5 


2.8 


1 1/2 


1 .7671 


30 


3.6 


4.0 


2 


3-1416 


53 


6.4 


7-i 


2 1/2 


4.9087 


8.2 


10. 


11 .0 


3 


7.0686 


II. 7 


14.2 


16.0 


3 1/2 


0.6211 


16.0 


19.5 


21 .6 


4 


12 .5660 


21 .0 


25-5 


28.2 


4 1/2 


15.9040 


26.7 


32.3 


36.0 


5 


196350 


32.7 


40.0 


44.0 



164 ARITHMETIC OF THE STEAM BOILER 

5. Each safety valve must have full-sized direct 
connection to the boiler, and full-sized escape pipe which 
shall be fitted with an open drain to prevent water 
lodging in the upper part of safety valve or escape pipe. 
When a boiler is fitted with two safety valves on one 
connection this connection to the boiler shall have a cross- 
sectional area equal to or greater than the combined area 
of the two safety valves. 

6. Safety valves having either the seat or disc of cast- 
iron shall not be used. 

7. The seats of all safety valves shall be inclined at an 
angle of forty-five (45) degrees to the center line of the 
spindle. 

8. A certificate of inspection shall not be issued on a 
boiler used for heating purposes exclusively, permitting 
the boiler to be operated at a pressure in excess of 
fifteen (15) pounds, if the boiler is provided with a device 
(safety valve) in accordance with the provision contained 
in section 78, chapter 102 of the Revised Laws, limiting 
the pressure carried to fifteen (15) pounds. 

9. Each boiler shall have a steam gage connected to the 
steam space of the boiler by a syphon, or equivalent 
device, sufficiently large to fill the gage tube with 
water, and in such manner that the steam gage cannot be 
shut off from the boiler except by a cock with T end, 
placed directly on the pipe under the steam gage. 

10. The dial of the steam gage shall be graduated to 
not less than one and one-half (1 1/2) times the maximum 
pressure allowed on the boiler. 

n. Each boiler shall be provided with a one-eighth 
(1/8) inch pipe size connection for attaching inspector's 



APPENDIX 165 

test gage when boiler is in service, so that the accuracy of 
the boiler steam gage can be ascertained, as required by 
section 3, chapter 465, Acts of 1907. 

12. Each boiler shall have one fusible plug, as required 
by rules (section 3) on fusible plugs. 

13. Each boiler shall have one water-glass, the bottom 
end of which shall be above the fusible plug and lowest 
safe water line. 

14. Each boiler shall have two or more gage cocks, 
located within the range of the water-glass, when the 
maximum pressure allowed does not exceed twenty-five 
(25) pounds per square inch. 

15. Each boiler shall have three or more gage cocks, 
located within the range of the water-glass, when the 
maximum pressure allowed exceeds twenty-five (25) 
pounds per square inch. 

16. Each steam outlet from boiler shall be fitted with a 
stop valve. 

17. When a stop valve is so located that water can 
accumulate, ample drains shall be provided. 

18. Each boiler shall have a feed pipe fitted with check 
valve, and also a stop valve between the check valve 
and the boiler, the feed water to discharge below the 
lowest safe water line. Means must be provided for 
feeding the boiler with w T ater when the maximum pressure 
allowed is carried on the boiler. 

19. Each boiler shall have a bottom blow-off pipe 
fitted with a stop valve or stop cock, and connected direct 
to the lowest water space of the boiler. 

20. Where a damper regulator is used, the boiler pres- 
sure pipe shall be taken from the steam space of the 



166 ARITHMETIC OF THE STEAM BOILER 

boiler, and shall be fitted with a stop valve or stop 
cock. 

21. Each boiler fitted with a Lamphrey Boiler Furnace 
Mouth Protector, or similar appendage, having valves 
on the pipes connecting same with the boiler, shall have 
these valves locked or sealed open, so that the locks or 
seals will require to be removed or broken to shut the 
valves. 

Annual Internal Inspections 

i. The owner or user of a steam boiler which requires 
annual inspection, internally and externally, by the boiler 
inspection department or by an insurance company, 
as provided by section i, chapter 465, Acts of 1907, 
shall prepare the boiler for inspection by cooling it down 
(blanking off connections to adjacent boilers if necessary), 
removing all soot and ashes from tubes, heads, shell, 
furnace and combustion chamber; drawing off the water; 
removing the handhole and manhole plates; removing 
the grate bars from internally fired boilers; and removing 
the steam gage for testing. 

2. If a boiler has not been properly cooled down, or 
otherwise prepared for inspection, the boiler inspector 
shall decline to inspect it, and he shall not issue a certifi- 
cate of inspection until efficient inspection has been made. 

3. In making the annual internal and external inspec- 
tion as provided by sections 1 and 4, chapter 465, 
Acts of 1907, the boiler inspector shall apply the hammer 
test to all internal and external parts of a boiler that are 
accessible. 



APPENDIX 167 

4. All proper measurements shall be taken by the boiler 
inspector, so that the maximum working pressure allowed 
on a boiler will conform to the rules relating to allowable 
pressures established by the Board of Boiler Rules; such 
measurements to be taken and calculations made before 
a hydrostatic pressure test is applied to a boiler. 

5. The steam gage of a boiler shall be tested and its 
readings compared with an accurate test gage, and if, 
in the judgment of the boiler inspector, the gage is not 
reliable he shall order it repaired or replaced. 

Annual External Inspections 

1 . The annual external inspection of a steam boiler, as 
provided for in section 3, chapter 465, Acts of 1907, 
should be made at or about six (6) months after the 
annual internal inspection, except in the case of a boiler 
that is in service a portion of the year only, in which 
case the annual external inspection shall be made during 
such period of service. 

2. The boiler inspector shall attach an accurate test 
gage to a boiler to note the pressure show r n by said 
test gage, and compare it with that shown by the boiler 
gage, ordering the boiler gage repaired or replaced if 
necessary. 

3. The boiler inspector shall see that the water-glass, 
gage cocks, water-column connections and w T ater blow- 
offs are free and clear; also that the safety valve raises 
freely from its seat. 

4. Fire doors, tube doors, and doors in settings shall 
be opened, to view as far as possible the fire surface. 



1 68 ARITHMETIC OF THE STEAM BOILER 

settings, tube ends, blow-off pipes and fusible plug, 
noting conditions and ordering changes or repairs if 
necessary. 

Hydrostatic Pressure Tests 

i . When a boiler is tested by hydrostatic pressure, the 
maximum pressure applied shall not exceed one and one- 
half (i 1/2) times the maximum working pressure allowed, 
except that tw T ice the maximum working pressure allow r ed 
may be applied on boilers permitted to carry tw T enty-five 
(25) pounds pressure per square inch or less, or on pipe 
boilers. 

2. When making annual inspections on boilers con- 
structed wholly of cast iron, or on pipe boilers, a hy- 
drostatic pressure test of not less than one and one-half 
(1 1/2) times and not more than twice the maximum 
working pressure allowed shall be applied. 

3. The boiler inspector, after applying a hydrostatic 
pressure test, shall thoroughly examine every accessible 
part of the boiler, both internal and external. 

TO DETERMINE MAXIMUM ALLOWABLE PRESSURE 



Formula: 
T.S.XtX% 



■■ maximum allowable working pressure, 



RXF.S. 
per square inch, in pounds. 
T.S. = Tensile strength of shell plates, in pounds. 
/ = minimum thickness of shell plates, in inches. 
% = efficiency of longitudinal joint. 



APPENDIX 169 

R = radius = one-half (1/2) the inside diameter of 
the outside course of the shell or drum. 
F.S.= lowest factor of safety allowed by these rules. 
When the tensile strength of steel or wrought iron 
shell plates is not known, it shall be taken as fifty-five 
thousand (55,000) pounds for steel, and forty-five 
thousand (45.000) pounds for wrought iron. 

Efficiency of Riveted Joints 

The efficiency that a unit of length of a riveted joint 
has to the same unit of length of solid plate shall be 
calculated as shown by the following examples: 
T.S. = tensile strength of plate, in pounds per square 
inch. 
/ = thickness of plate, in inches. 
b = thickness of butt strap, in inches. 
P = pitch of rivets, in inches, on row having greatest 

pitch. 
d = diameter of rivet after driving, in inches. 
a = cross-sectional area of rivet after driving, in 

square inches. 
5 = strength of rivet in single shear. 
5 = strength of rivet in double shear. 
C = crushing strength of mild steel. 
(Note. — "C" applies only to boilers constructed after 
February 5, 1910.) 

n = number of rivets in single shear in a unit of 

length of joint. 
N = number of rivets in double shear in a unit 
of length of joint. 






I70 



ARITHMETIC OF THE STEAM BOILER 



Lap, Single-riveted. 

longitudinal or circumferential 

Example. 

A ^strength of solid plate =PXtXT.S. 

B = strength of solid plate between rivet holes 

= (P-d)XtXT.S. 
C = shearing strength of one rivet in single shear 

= nXsXa. 




Fig. i. — Single-riveted lap joint. 



D = crushing strength of plate in front of one (i) 
rivet =dXtXc 
Divide B, C, or D (whichever is the least) by A, and 
the quotient will be the efficiency of a single-riveted lap 
joint. 

7\S.=S5,ooolb. 

t = i/4 in. =.25 in. 
d = 11/16 in. =.6875 in. 
a= .3712 sq. in. 
5 =42,000 lb. 
c =95, 000 lb. 
A =i.625X. 25X55,000 = 22,343. 



APPENDIX 



171 



5 = (1.625 -.6875). 25X55,000 = 12. 890. 
C = iX42,oooX.37i2 = 15.590. 
D = . 6875 X. 25X9S.000 =16,328. 

12890 QB) u . 

22343 ( A ) = -576, efficiency of joint. 

(See Fig. 1 of this group.) 

Lap, Double-riveted. 

longitudinal or circumferential 
Strength of solid plate = PXtX T.S. = A. 



W-pA 




6 — ^ — — §-} 



Fig. 2. — Double- riveted lap joint, 

Strength of plate between rivet holes= (P—d)tXT.S. 
= B. 

Shearing strength of two (2) rivets in single shear 
= nsa = C. 

Divide B or C (whichever is the least) by A , and the 
quotient will be the efficiency of a double-riveted lap 
joint. 

T.S. = 55,000 lb. 

*=5/i6 in. = .3125 in. 
P=2 7/8 in. = 2.875 i n - 



172 ARITHMETIC OF THE STEAM BOILER 

d = 3/4 in. = .75 in. 

a= .4418 sq. in. 

5 = 42,000 

A = 2. 875X. 3125X55.000 = 49,414. 
B = (2. 875 -.75) X. 3125X55,000 = 36, 523- 

C = 2 X. 4418X42,000 = 37,111. 

36523 (B) „. . ... 

ttt = -7^9, emiciency of joint. 

49414 UJ MV J J 

(See Fig. 2 of this group.) 



Butt, Double-riveted. 

butt and double-strap joint 
,4=strength of solid plate=PX*XT.S. 




Fig. 3. — Double- rive ted, double butt-strapped joint. 

B = strength of plate between rivet holes in 
the outer row = (P-d)tXT.S. 

C = shearing strength of two (2) rivets in 
double shear; plus the shearing strength 
of one (1) rivet in single shear = NXS 
Xa+nXsXa. 



APPENDIX 173 

D= strength of plate between rivet holes in 

the second row, plus the shearing strength 

of one (1) rivet in single shear in the 

outer row=(P-2d)tXT.S.+nXsXa. 

E= strength of plate between rivet holes in 

the second row, plus the crushing 

strength of butt strap in front of one 

(1) rivet in the outer row = (P — 2d) 

tXT.S.+dXbXc. 

F = crushing strength of plate in front of 

two (2) rivets, plus the crushing strength 

of butt strap in front of one (1) rivet = 

NXdXtXc+nXdXbXc. 

G = crushing strength of plate in front of two (2) 

rivets, plus the shearing strength of one (1) 

rivet in single shear = NXdXtXc+nXsX a. 

Divide B, C, D, £, F, or G (whichever is the least) 

by A, and the quotient will be the efficiency of a butt and 

double-strap joint, double-riveted. (See Fig. 3 of this 

group.) 

T.S. = 55,000 lb. a = .6013 sq. in. 

t = s/S in. = -375 in. 5 = 42,000 lb. 

6 = 5/16 in. = .3125 in. 5 = 78,000 lb. 

P = 4 7/8 in. =4.875 in. £ = 95,000 lb. 

Number of rivets in single shear in a unit of length of 
joint=i. 

Number of rivets in double shear in a unit of length of 
joint =2. 

^l=4.875X.375X55^oo= 100,547. 

B = (4.875 — .875) .375X55^00 = 82,500. 



174 



ARITHMETIC OF THE STEAM BOILER 



C = 2X 78,000 X. 6013 + iX42,oooX. 6013 = 

119,057. 
D= (4.87S -2X.875).375X55> 000 + I X42,oooX 

.6013 = 89,708. 

E= (4-875- 2X.87s).37sX5S»oo°+-875X.3i2S 
X95 ? ooo = 9o,429. 

F = 2X.87sX.37SX9S ? 000 +- 8 75X.3i2SX9S ? 000 

= 88,320. 
G=2X.87sX.37SX9S,ooo+iX4 2 ) 00 °X.6oi3 = 

87,599. 

82500(^8) rr • r • • 4 

— - — V-v = .82o, emciency of joint. 
100547^) 

(See Fig. 3 in this group.) 
Butt, Triple-riveted, 
butt and double-strap joint 
A =strength of solid plate =PXtXT.S. 




Fig. 4. — Triple- riveted, double butt-strapped joint. 

B = strength of plate between rivet holes in the 
outer row= (P-d)tX T.S. 

C = shearing strength of four (4) rivets in double 
shear, plus the shearing strength of one (1) 
rivet in single shear = NXSXa+nXsXa. 



APPENDIX 175 

D = strength of plate between the rivet holes in 
the second row, plus the shearing strength of 
one (1) rivet in single shear in the outer row 
= (P-2d)tXT.S.+nXsXa. 

E = strength of plate between rivet holes in the 

second row, plus the crushing strength of butt 

strap in front of one (1) rivet in the outer row 

= (P-2d)tXT.S.+dXbXc. 

F = crushing strength of plate in front of four (4 J 

rivets, plus the crushing strength of butt strap 

in front of one (1) rivet = XXdXtXc+nXdX 

bXc 

G = crushing strength of plate in front of four (4) 

rivets, plus the shearing strength of one (1) 

rivet in single shear = XXdXtXc+nXsXa. 

Divide B, C, D, E, F, or G (whichever is the least) by 

A j and the quotient will be the efficiency of a butt and 

double-strap joint, triple riveted. (See Fig. 4 of this 

group.) 

T.S. = 55,000 lb. # = .5185 sq. in. 

t = $/8 in. = .375 in. 5 = 42,000 lb. 

6 = 5/16 in. = .3125 in. 5 = 78,000 lb. 

F = 6 1/2 in. = 6.5 in. £ = 95,000 lb. 
d= 13/16 in. = .8125 

Number of rivets in single shear in a unit of length of 
joint = 1. 

Number of rivets in double shear in a unit of length of 
joint = 4. 

-4=6.5X.375X55> 000=I 34,o62. 
^ = .( 6 -5-- 8l2 5)-37SX 55,000 =117,304. 



176 ARITHMETIC OF THE STEAM BOILER 

C = 4 X 78,000 X. 5185 + 1 X 42,000 X-5i85 = 

183,549- 
£ = (6.5-2 X.8i25). 3 75X55>°oo+iX 42,000 X 

.5185 = 122,323. 
£ = (6.5-2 X.8i25). 3 75 X 55,000+. 8125 X.3125 

X95,ooo= 124,667. 

£ = 4X.8i25X.375X95>°oo+iX.8i25X.3i25X 
95,000=139,902. 

G = 4 X.8125 X.375X95,ooo+iX 42,000 X. 5185 

= i37 ; SS8. 

i34o?fe) == - 87S ' efficienc y of i° int - 

(See Fig. 4 in this group.) 

Butt, Quadruple-riveted. 

butt and double-strap joint, quadruple -riveted 

A = Strength of solid plate =PXtXT.S. 

B = Strength of plate between rivet holes in the 

outer row =(P-d)tXT.S. 
C = shearing strength of eight (8) rivets in 

double shear, plus the shearing strength 

of three (3) rivets in single shear = NXSX 

a-\-nXsXa. 
D = strength of plate between rivet holes in the 

second row, plus the shearing strength of 

one (1) rivet in single shear in the outer row 

= (P-2d)tXT.S.+nXsXa. 
E = strength of plate between rivet holes in the 

third row, plus the shearing strength of two 



APPENDIX 



177 



(2) rivets in the second row in single shear 
and one (1) rivet in single shear in the outer 
row=(P^4d)tXT.S.+nXsXa. 

F = strength of plate between rivet holes in the 
second row, plus the crushing strength of 
butt strap in front of one (1) rivet in the 
outer row=(P-2d)tXT.S.+dXbXc 

G = strength of plate between rivet holes in 




Fig. 5. — Quadruple-riveted, double butt-strapped joint. 

the third row, plus the crushing strength of 
butt strap in front of two (2) rivets in the 
second row and one (1) rivet in the outer 
row=(P-4d)tXT.S.+nXdXbXc. 

H = crushing strength of plate in front of eight 
(8) rivets, plus the crushing strength of 
butt strap in front of three (3) rivets = 
NXdXtXc+nXdXbXc 

1 = crushing strength of plate in front of eight 
(8) rivets, plus the shearing strength of 
two (2) rivets in the second row and one 



178 ARITHMETIC OF THE STEAM BOILER 

(1) rivet in the outer row, in single shear = 
NXdXtXc+nXsXa. 

Divide B, C, D, E, F, G, H, or / (whichever is the least) 
by A, and the quotient will be the efficiency of a butt 
and double-strap joint, quadruple-riveted. 

T.S. = 55,000 lb. 

t= 1/2 in. = .5 in. 

6 = 7/16 in. = .4375 
P=iS in. 

d= 15/16 in. = .9375 in. 

a = .6903 sq. in. 

5 = 42,000 lb. 
5 = 78,000 lb. 

c = 95,000 lb. 

Number of rivets in single shear in unit of length of 
joint = 3. 

Number of rivets in double shear in unit of length of 
joint = 8. 

^ = i5X.5X55> 000 = 4i2,5oo. 

£ = (i5--9375)-5X55>°°o = 386,7i8. 

C = 8 X 78,000 X.6903 + 3 X 42,000 X.6903 = 

5i7,723- 
#=(15-2 X.937S)-S X 55,ooo+ 1 X 42,000 X 

•69°3 = 389>93°- 
£=(15-4 X .9375)-5 X 55,000 + 3 X 42,000 X 

.6903 = 396,353. 
^ = (15-2 X.9375)-5X 55>°°°+-9375 X-4375 X 

95,000 = 399,902. 



APPENDIX 179 

G=(i5-4 X .937S)-S X 55>°oo + 3 X -9375 X 

•4375X95> 000 = 426,269. 
H = 8X -937S X.5 X 95,000 + 3 X-937SX.437SX 

95,000 = 473,145. 
7 = 8 X -9375 X .5 X 95>°°° + 3 X 42,000 X 

.6903 = 443,229. 

386718(5) ffi . , . . ■ 

4i25oo04r- 937 ' efikienCy ° f JOmt * 

(See Fig. 5 of this group.) 

Bumped Heads 

The minimum thickness of a convex head shall be 
determined by the following formula: 

RXF.S.XP 
T.S. 

The minimum thickness of a concave head shall be 
determined by the following formula: 

RXF.S.XP 

.6(T.S.) 

R = one-half the radius to which the head is 
bumped. 
F.S. = 5 = factor of safety. 

P = working pressure, in pounds per square inch, 
for which the boiler is designed. 
T.S. = tensile strength, in pounds per square inch, 
stamped on the head by the manufacturer. 
t = thickness of the head in inches. 
When a convex or a concave head has a manhole 
13 



180 ARITHMETIC OF THE STEAM BOILER 

opening, the thickness as found by the preceding for- 
mulas shall be increased by not less than one-eighth (1/8) 
inch. 

FORMULA TO FIND AREA OF SEGMENT OF CIRCLE TO 
BE BRACED 



~ a / jj— .608 = area in square inches. 

H = distance from tubes to shell, minus five (5) 

inches. 
R = radius of boiler, minus three (3) inches. 

FORMULA FOR DIAMETER OF STAY BOLTS AT BOTTOM 
OF THREAD 

D — (PX 1.732) =d, or for 12 threads per inch, 
£>-(.o8333Xi.73 2 )=^ then 

D = diameter of stay bolt over the threads. 

P = pitch of threads = 1/ 1 2 = .08333. 

d = diameter of stay bolt at bottom of threads. 
1.732 = a constant. 

When U. S. threads are used, the formula becomes: Z) — 
(PX1.732X.75W- 

FORMULA FOR CAST-IRON NOZZLES 

The minimum thickness of cast-iron nozzles shall be 
determined by the following formula: 

Pdf , 



APPENDIX 181 

P = working pressure in pounds per square inch. 

d = inside diameter of nozzle in inches. 

/= factor of safety = 12. 

5 = ultimate tensile strength of cast iron, not less 

than eighteen thousand (18,000) pounds per 

square inch. 
.5 = a constant. 
t = thickness of nozzle in inches. 

Maximum Pressure on Boiler Shells 

The maximum pressure to be allowed on a steel or 
wrought-iron shell or drum shall be determined from the 
minimum thickness of the shell plates, the lowest 
tensile strength stamped on the plates by the plate 
manufacturer, the efficiency of the longitudinal joint or 
ligament between the tube holes, whichever is the least, 
the inside diameter of the outside course, and a factor of 
safety of not less than five (5), the formula being: 

T.S. X^X % 
7?v7? y — = maximum allowable working pressure per 

square inch in pounds. 

T. S. = tensile strength of shell plates in pounds. 
t = minimum thickness of shell plates in inches. 
% = efficiency of longitudinal joint or ligament be- 
tween tube holes, whichever is the least. 
R = radius = one-half (1/2) the inside diameter of the 
outside course of the shell or drum. 
F.5. = 5, the lowest factor of safety allowed on boilers 
installed after May 1, 1908. 



182 



ARITHMETIC OF THE STEAM BOILER 



The method of determining the efficiency of the 
longitudinal joint has already been explained and illus- 
trated. To find the efficiency of ligaments, the following 
formulas are to be employed. 

Efficiency of Ligaments 

When a shell or drum is drilled for tube holes in a line 
parallel to the axis of the shell or drum, the efficiency 
of the ligament between the tube holes shall be deter- 

-(£-&- Q Ct) (fr d> 



<-5 1 4-> 



<— &£->*— VA-* 



<r-&i 



-&M-» 



-5*4 



,-5;£-> 



^ ) Q Cp Cp Cp Cp C p-^)- 

Longitudinal Line >. 

Fig. i. — Diagram for calculating the efficiency of ligament. 

mined as follows: (a) when the pitch of the tube holes 
on every row is equal the formula is: 

p d 

— - — = efficiency of ligament. 



p = pitch of tube holes in inches. 
d = diameter of tube holes in inches. 

Example. — Pitch of tube holes in the drum of a water-tube 
boiler = 5 1/4 in. = 5.25 in. Diameter of tube holes = 3 1/4 in. = 
3.25 in. 



P~d 5-25-3-25, 



: .38, efficiency of ligament. 



5-25 
(See Fig. 1 of this group.) 



APPENDIX 



183 



(b) when the pitch of the tube holes on any one row is unequal, 
the formula is: 



P-na 



= efficiency of ligament. 



■e 



e-©-e 



^._ 5 4-/W--6^-^-54i^--64f , ->U--544^[<---6^-^--54- 



h— -1-2^— h 

Longitudina l Line 

Fig. 2. — Diagram for calculating the efficiency of ligament. 



P = unit length of ligament in inches. 
n = number of tube holes in length, P. 
d = diameter of tube holes in inches. 






-29 H- 
J Longitudinal Line 

Fig. 3. — Diagram for calculating the efficiency of ligament. 



Example. — 

P — nd _i2 — 2X3. 25 



= .458, efficiency of ligament. 



(See Fig. 2 of this group.) 
Example. — 

P—nd _ 2g. 25-5X3.25 



.444, efficiency of ligament. 
(See Fig. 3 of this group.) 



1 84 ARITHMETIC OF THE STEAM BOILER 

When a shell or drum is drilled for tube holes in a line diagonal to 
the axis of the shell or drum, the efficiency of the ligament between 
the tube holes shall be determined as follows: 

P-d 

— = efficiency of ligament. 

P 

P = diagonal pitch of tube holes in inches. 
d = diameter of tube holes in inches. 
p = distance between rows of tubes longitudinally. 






{ p'cfo ($> <fe- 



Girth Line_ 

Fig. 4. — Diagram for calculating the efficiency of ligament. 

Example. — Diagonal pitch of tube holes in a drum of a water- 
tube boiler = 6.42 in. 
Diameter of tube holes =4 in. 
Distance between rows of tubes, longitudinally = 5.75 in. 

P-d 6. 42-4 

— — = = .42, efficiency of ligament. 

P 5-75 

(See Fig. 4 of this group.) 

When a flat-head has a manhole opening, the flange 
of which is formed from the solid sheet and turned inward 
to a depth of not less than twice the thickness of the 
head, an area two (2) inches wide all around the manhole 
opening, as shown in the figure, may be deducted from 
the total area of head, including manhole opening, to be 
stayed. 



APPENDIX 



185 



Example. — To find an area 2 in. wide all around a 11 in. X 15 in. 
manhole, 

15 in.Xi9 in. X. 7854 = 224 (nearly) sq. in. 
11 in.Xi5 in. X. 7854 = 130 (nearly) sq. in. 
And 224 — 130 = 94 sq. in. 

Therefore, if the area to be stayed on the rear head, 
below the tubes, of a seventy-two (72) inch horizontal re- 
turn tubular boiler is 374 sq. in., the area to be stayed on 
front head, below the tubes, of this boiler, would be 374 — 
94 = 280 sq. in. 



k AAA m^-'^mm fT Yfhfh / 



0B^ 



Fig. 1. — Diagram showing area of head to be braced. 

A segment of a head of a horizontal return tubular, 
locomotive, Scotch or similar type shall be stayed by 
welded or weldless mild steel or wrought iron, head to 
head or through, diagonal or crow-foot stays, except a 
horizontal return tubular boiler, as otherwise provided 
for. 

The area of a segment of a head to be stayed shall 
be the area enclosed by lines drawn three (3) inches from 



1 86 ARITHMETIC OF THE STEAM BOILER 

the shell and two (2) inches from the tubes, as shown in 
Figs. 1 and 2. 

When the shell of a horizontal return tubular boiler 
does not exceed thirty-six (36) inches in diameter, and 
is designed for a maximum working pressure not to exceed 
one hundred (100) pounds per square inch, the segment 
of the head above the tubes may be stayed by steel 
angles, or Tee bars, the formula being: 




Fig. 2. — Diagram showing area of head to be braced. 







J -=M 

y 




/= 


= fiber stress = 


= 16,000 


lb. 


/= 


= moment of inertia = 


12 



J A = height of beam in inches. 
\b = thickness of beam in inches. 
y = distance of most strained fiber = h-S- 2. 
M = bending moment of beam. 



APPENDIX 



187 



load = 



maximum bending moment for a uniform 
WL 



W = weight to be supported in pounds. 
L = length of beam in inches. 



Example. — When steel angles are used, the head of a horizontal 
return tubular boiler thirty (30) inches in diameter, designed for 




Fig. 1. — Diagram of steel angles bracing. 



one hundred (100) pounds working pressure, shall be stayed by 
two (2) four and one-half by three by three-eighths (41/2X3X3/8) 
inch steel angles, as shown in the figure, or by other sized commercial 
steel angles, the resistance of which shall be equal to or greater than 
the maximum bending moment. 

Distance from tubes to shell = 13 1/2 in. 



i88 



ARITHMETIC OF THE STEAM BOILER 



Area to be stayed = 143 sq. in. 
Load at 100 lb. pressure = 14.300 lb. 

WL 14300X21. 

~g- = g = 37,54o lb. 

Moment of inertia = / = i/i2X4.5 3 X3/8 = 2.85. 
^=4.5-^-2 = 2.25. 




Fig. 2. — Diagram of steel angles bracing. 



I 1 -, , 16000X2.85 ■ 

— =M = — — = 20,266 lb. for one angle. 

y 2.25 

Resistance of one angle = 20,266 lb. 
Resistance of two angles = 40,532 lb. 

When steel angles are used, the head of a boiler, 
thirty-six (36) inches in diameter, designed for one 
hundred (100) pounds working pressure, shall be stayed 



APPENDIX 189 

by two (2) six by three and one-half by one-half (6X 
3 1/2 X 1/2) inch steel angles, as shown in the figure 
or by other sized commercial steel angles the resistance 
of which shall be equal to or greater than the maximum 
bending moment. 

Distance from tubes to shell=i5 1/2 in. 

Area to be stayed=22o sq. in. 

Load at 100 lb. pressure = 22,000 lb. 

WL 22000X27 __ ^_ . . 

-0~=— — 5 =74,250 lb. Moment of inertia = 

o o 

Z=i/i2X6 3 Xi/2 = 9. 

fl __ 16000X9 » r , 

— =M=— —=48,000 lb. for one angle. 

y 3 

Resistance of one angle = 48,000 lb. 
Resistance of two angles = 96,000 lb. 

PITCH OF STAY BOLTS ON FURNACE SHEETS 

The longitudinal pitch between stay bolts on the fur- 
nace sheet of an internally fired boiler, in which the ex- 
ternal diameter of the furnace is thirty-eight (38) inches 
or less, except a corrugated furnace or a furnace strength- 
ened by an Adamson ring or equivalent, shall not exceed 
that given by the following formula: 






190 ARITHMETIC OF THE STEAM BOILER 

L = longitudinal pitch of stay bolts, in inches, or one- 
half the height of furnace when only one circumfer- 
ential row of stay bolts is required. 
C = a constant = no. 

t = thickness of furnace sheet, in thirty-seconds of an inch . 
P = working pressure per square inch in pounds. 
d = external diameter of furnace in inches. 



TABLE I.— AREAS AND CIRCUMFERENCES 


OF 


CIRCLES FROM 






I TO IOO 


Dia. 


Area 


Circum. 


Dia. 


Area 


Circum. 


Dia. 


Area j Circum. 


3 \ 


0.00077 


O.098175 


2 


3.1416 


6.28319 


5 l 


I9-635 


15.7080 


3 
^4 


0.00173 


O.147262 


1 
1 6 


3-34io 


6.47953 


1 

T6 


20.129 


15-9043 


1 

Tg 


0.00307 


O.196350 


1 
8 


3-5466 


6.67588 


1 

8" 


20.629 


16.1007 


3 

32 


0.00690 


O.294524 


A 


3-75 8 3 


6.87223 


3 

1 6 


21.135 


16.2970 


i 


0.01227 


O.392699 


1 

4 


3.9761 


7.06858 


1 
4" 


21.648 


16.4934 


A 


0.01917 


O.490874 


5 
16 


4.2000 


7.26493 


5 

T6 


22.166 


16.6897 


A 


0.02761 


O.589049 


t 


4.4301 


7.46128 


3 

8 


22.691 


16.8861 


3"2~ 


0.03758 


O.687223 


A 


4.6664 


7^37^3 


16 


23.221 


17.0824 


1 


O.04909 


O.785398 


i 


4.9087 


7.85398 


1 
2 


23758 


17.2788 


9 
32 


0.06213 


O.883573 


9 

16 


5-I572 


8.05033 


9 

1 6 


24.301 


I7475I 


5 
16 


0.07670 


O.981748 


5 

8 


5-4II9 


8.24,668 


f 


24.850 


17.6715 


11 
32 


0.09281 


I.07992 


1 1 
16 


5.6727 


844303 


1 1 

16 


25.406 


17.8678 


3 

8 


0.11045 


I.17810 


3 
4 


5-9396 


8.63938 


3 
4 


25.967 


18.0642 


M 


0.12962 


I.27627 


13 
16 


6.2126 


8.83573 


13 

16 


26.535 


18.2605 


JL- 

16 


O.I5033 


1-37445 


7 
8 


6.4918 


9.03208 


7 
8 


27.109 


18.4569 


M 


0.17257 


1.47262 


15 
16 


6.7771 


9.22843 


1 5 
16 


27.688 


18.6532 


1 
2 


O.19635 


1.57080 


3 


7.0686 


9.42478 


6 


28.274 


18.8496 


17 
32 


O.22166 


1.66897 


1 

1 6 


7.3662 


9.62113 


i 


29.465 


19.2423 


9 
T6 


0.24850 


1.76715 


i 


7.6699 


9.81748 


i 


30.680 


19.6350 


19 
32 


0.27688 


1.86532 


3 

T6 


7.9798 


10.0138 


f 


3 J -9i9 


20.0277 


1 


0.30680 


1.96350 


i 


8.2958 


10.2102 


1 
2 


33-^3 


20.4204 


21 
32 


0.33824 


2.06167 


A 


8.6179 


10.4065 


5 

8 


34472 


20.8131 


a 


O.37122 


2.15984 


I 


8.9462 


10.6029 


3 
4 


35785 


21.2058 


If 


O.40574 


2.25802 


A 


9.2806 


10.7992 


i 


37.122 


21.5984 


1 


O.44179 


2.35619 


i 


9.6211 


10.9956 


7 


38485 


21.9911 


2.5 
32 


0.47937 


2-45437 


9 
16 


9.9678 


11.1919 


1 

8 


39-871 


22.3838 


1 3 
T6 


0.51849 


2 -55254 


5 

8 


10.321 


11.3883 


1 

4 


41.282 


22.7765 


2.1 
32 


0.559I4 


2.65072 


tt 


10.680 


11.5846 


1 


42.718 


23.1692 


i 


0.60132 


2.74889 


1 


11.045 


11.7810 


i 


44.179 


23.5619 


If 


0.64504 


2.84707 


H 


11.416 


11.9773 


f 


45.664 


23.9546 


1 5 
T6 


O.69029 


2.94524 


I 


JI -793 


12.1737 


f 


47.173 


24.3473 


3 1 
32 


O.73708 


3-04342 


15 

16 


12.177 


12.3700 


1 


48.707 


24.7400 


I 


O.78540 


3-I4I59 


4 


12.566 


12.5664 


8 


50.265 


25.1327 


1 

16 


O.88664 


3-33794 


A 


12.962 


12.7627 


i 


51.849 


25.5224 


i 


0.99402 


3-53429 


1 

8 


13-364 


12,9591 


1 
4 


53-456 


25.9181 


A 


I.1075 


3-73064 


A 


!3-77 2 


I 3- I 554 


I 


55.o88 


26.3108 


1 

4 


I.2272 


3.92699 


i 


14.186 


i3-35i8 


1 


56.745 


26.7035 


A 


I-3530 


4-12334 


5 

XT 


14.607 


i3-548i 


5 

8 


58.426 


27.0962 


1 


1.4849 


4.31969 


I 


J 5-o33 


J 3-7445 


1 


60.132 


27.4889 


A 


1.6230 


4.51604 


A 


15.466 


13.9408 


i 


61.862 


27.8816 


1 


1. 7671 


4.71239 


i 


15.904 


14.1372 


9 


63.617 


28.2743 


A 


J-9I75 - 


4.90874 


9 
16 


16.349 


J 4-3335 


i 


65.397 


28.6670 


f 


2.0739 


5.10509 


1 


16.800 


.14.5299 


1 
4 


67.201 


29.0597 


16 


2.2365 


5-3oi44 


tt 


I7-257 


14.7262 


f 


69x29 


29.4524 


i 


2.4053 


5-49779 


f 


17.721 


14.9226 


i 


70.882 


29.8451 


1 3 
T6 


2.5802 


5.69414 


H 


18.190 


15.1189 


i 


72.760 


30.2378 


A 


2.7612 


5.89049 


7 
8 


18.665 


J 5-3i53 


J 


74.662 


30-6305 


To 


2.9483 


6.08684 


15 

T6 


19.147 


i5-5ii6 


I 


76.589 


31.0232 



191 



TABLE I.— AREAS AND CIRCUMFERENCES OF CIRCLES FROM 
i TO ioo (Continued) 



Dia. 


Area 


Circum. 


Dia. 


Area 


Circum. 


Dia. 


Area 


Circum. 


IO 


78.540 


3I-4I59 


16 


201.06 


50-2655 


22 


380.13 


69.1150 


i 


80.516 


31.8086 


i 


204.22 


50.6582 


1 
s 


384.46 


69.5077 


l 

4 


82.516 


32.2013 


1 

4 


207.39 


51.0509 


1 

4 


388.82 


69.9004 


f 


84.541 


3 2 -5940 


3 

8 


210.60 


5I-4436 


3 

8 


393.20 


70.2931 


i 


86.590 


32.9867 


1 
2 


213.82 


51.8363 


1 
2 


397-6l 


70.6858 


5 

8 


88.664 


33-3794 


1 


2 1 7 .08 


52.2290 


5 

. 8 


402.04 


71.0785 


f 


90.763 


33-7721 


3 
4 


22O.35 


52.6217 


3 
4 


406.49 


71.4712 


7 
8 


92.886 


34.1648 


7 
8 


223.65 


53-OI44 


8 


410.97 


71.8639 


II 


95-033 


34.5575 


17 


226.98 


53-407I 


23 


415.48 


72.2566 


1 

8 


97- 2 05 


34-9502 


1 

8 


230-33 


53-7998 


i 


420.00 


72.6493 


1 
4 


99.402 


35-3429 


i 


233-7I 


54.1925 


1 
4 


424.56 


73.0420 


3 

8 


101.62 


35-7350 


3 

8 


237.IO 


54o852 


1 


429.13 


73-4347 


1 

2 


103.87 


36.1283 


4 


240.53 


54.9779 


1 

2 


433-74 


73.8274 


1 


106.14 


36.5210 


5 

8 


243.98 


55.3706 


5 

8 


438.36 


74.2201 


3 
4 


108.43 


36.9137 


3 
4 


247-45 


557633 


3 
4 


443.01 


74.6128 


7 
8 


110.75 


37-3o64 


7 
8 


250.95 


56.1560 


1 
8 


447.69 


75.0055 


12 


113.10 


37.6991 


l8 


25447 


56.5487 


24 


452.39 


75.3982 


i 


115-47 


38.0918 


1 
8 


258.02 


56.9414 


1 
8 


457-n 


75.7009 


i 

4 


117.86 


38.4845 


1 
4 


261.59 


57.3341 


1 


461.86 


76.1836 


§ 


120.28 


38.8772 


i 


265.18 


57.7268 


1 


466.64 


76.5783 


4 


122.72 


39.2699 


4 


268.80 


58.1195 


i 


471.44 


76.9690 


f 


125.19 


39.6626 


5 

8 


272.45 


58.5122 


5 

8 


476.26 


77.3617 


4 


127.68 


40.0553 


3 

4 


276.12 


58.9049 


f 


481. 11 


77-7544 


i 


130.19 


40.4480 


i 


279.81 


59.2976 


I 


485.98 


78.1471 


13 


132.73 


40.8407 


19 


283.53 


59.6903 


25 


490.87 


78.5398 


4 


J 35-3o 


41.2334 


4 


287.27 


60.0830 


i 


495-79 


78.9325 


J 


137.89 


41.6261 


i 


291.04 


60.4757 


1 

4 


500.74 


79.3252 


I 


140.50 


42.0188 


3 

8 


294.83 


60.8684 


| 


505. 7 1 


79.7179 


4 


I43-I4 


42.4115 


i 


298.65 


61.2611 


i 


510.71 


80.1105 


f 


145.80 


42.8042 


5 

8 


302.49 


61.6538 


5 

8 


51572 


80.5033 


f 


148.49 


43.1969 


3 
4 


306.35 


62.0465 


3 
4 


520.77 


80.8960 


7 
8 


151.20 


43-5896 


7 
8 


310.24 


62.4392 


7 

8 


525-84 


81.2887 


14 


153.94 


43.9823 


20 


314.16 


62.8319 


26 


530.93 


81.6814 


J 


156.70 


44-375° 


1 

8 


318.10 


63.2246 


4 


536.05 


82.0741 


J 


15948 


447677 


1 
4 


322.06 


63.6173 


1 

4 


541.19 


82.4668 


1 


162.30 


45.1604 


I 


326.05 


64.OIOO 


1 


546.35 


82.8595 


i 


l6 5-i3 


45-5531 


4 


330.06 


64.4026 


i 


55 I -55 


83.2522 


1 


167.99 


45-9458 


f 


334-IO 


64.7953 


5. 

8 


556.76 


83.6449 


f 


170.87 


46.3385 


3 
4 


338.16 


65.1880 


3 
4 


562.00 


84.0376 


i 


173.78 


46.7312 


7 
8 


342.25 


65.5807 


7 
8 


567.27 


84 4303 


J 5 X 


176.71 


47.1239 


21 


346.36 


65.9734 


27 


572.56 


84.8230 


i 


179.67 


47-5166 


1 

8 


350-5° 


66.3661 


i 


577.87 


85-2157 


i 


182.65 


47.9093 


1 


354-66 


66.7588 


1 
4 


583-21 


85.6084 


I 


185.66 


48.3020 


I 

8 


358.84 


67-i5 I 5 


f 


588.57 


86.0011 


4 


188.69 


48.6947 


i 


363.05 


67.5442 


J 


593-96 


86.3938 


f 


I9I-75 


49.0874 


1 


367.28 


67.9369 


5 


599.37 


86.7865 


I 


194.83 


49.4801 


I 


37L54 


68.3296 


1 


604.81 


87.1792 


i 


T97.93 1 


49.8728 


I 


375.83 


68.7223 


i 


610.27 


87-5719 



TQ2 



TABLE I.— AREAS AND CIRCUMFERENXES OF CIRCLES FROM 
i TO ioo (Continued) 



Dia. Area 


Circum. 


Dia. 


Area 


Circum. 


Dia 


Area 


Circum. 


28 


6I5-75 


87.9646 


34 


907.02 


106.814 


40 


1256.6 


125.664 


1 
8 


621.26 


83-3573 


1 

8 


914.61 


107.207 


1 


1264.5 


126.056 


i 626.80 


88.7500 


1 
4 


921.32 


107.600 


1 
4 


1272.4 


126.449 


I 632.36 


89.1427 




928.06 


107.992 


1 


i 1280.3 


126.842 


i 637.94 


89-5354 


1 


934.82 


108.385 


1 

• 


1288.2 


I2 7-235 


f 


I 643o5 


89.9281 


i 


941.61 


108.788 


5 

8 


. 1296.2 


127.627 


3 

4 


649.18 


90.3208 


3 
4 


948.42 


109.170 


f 


1304.2 


128.020 


7 
8 


656.84 


90.7I35 


i 


955-25 


109.563 


1 


1312.2 


128.413 


29 


660.52 


91.1062 


35 


962.11 


109.956 


41 


!32o.3 


128.805 


1 


666.23 


91.4989 


1 

8 


969.00 


110.348 


1 
8 


J 328.3 


129.198 


1 
4 


67I.96 


9 1. 89 1 6 


1 
4 


975-91 


no. 741 


1 
4 


!336-4 


129.591 


§ 


677.71 


92.2843 


3 

8 


982.84 


III. 134 


3 

8 


1344.5 


129.993 


i 


683.49 


92.6770 


i 


989.80 


III.527 


1 

2 


J352.7 


^Z ^^ 


5 

8 


689.3O 


93.0697 


5 

s 


996.78 


1 1 1 .9 1 9 


5 

8 


1360.8 


130.769 


3 
4 


695-I.3 


93.4624 


3 
4 


1003.8 


112. 312 


3 

4 


1369.0 


131. 161 


s 


7OO.98 


93.8551 


7 
8 


1010.8 


112.705 


7 
8 


!377.2 


I 3 I -554 


3° 


706.86 


94.2478 


36 


1017.9 


113.097 


42 


i3 8 54 


I 3 I -947 


1 

8 


712.76 


94.6405 


i 


1025.0 


113.490 


1 

8 


13937 


132.340 


1 
4 


718.69 


95-033 2 


1 
4 


1032. 1 


113.883 


1 
4 


1402.0 


132.732 


1 


724.64 


954259 


3 

8 


1039.2 


114.275 


3 

8 


1410.3 


I 33-i25 


4 


730.62 


95.8186 


1 

2 


1046.3 


114.668 


1 

2 


1418.6 


I 33-5 l8 


i 


736.62 


96.2113 


f 


IQ 53-5 


115.061 


f 


1427.0 


!33-9io 


3 
4 


742.64 


96.6040 


3 

4 


1060.7 


1 15-454 


I 


14354 


J 34-303 


7 

8 


748.69 


96.9967 


7. 
8 


1068.0 


115.846 


1 


1443.8 


134.696 


3 1 


754-77 


97-3894 


37 


1075.2 


116.239 


43 


1452.2 


135.088 


1 

8 


760.87 


97.7821 


1 

8 


1082.5 


116.632 


1 

8 


1460.7 


135481 


i 


766.99 


98.1748 


i 


1089.8 


117.024 


1 
4 


1469. 1 


J 35-874 


1 


773-14 


98.5675 


3 

8 


1097. 1 


117.417 


I 


1477.6 


136.267 


4 


779-3 1 


98.9602 


1 
2 


1104.5 


117.810 


J 


1486.2 


136.659 


1 


785.51 


99.3529 


5 

8 


1111.8 


118.202 


5 

8 


1494.7 


!37-o52 


3 

4 


791-73 


99.7456 


3 
4 


1119.2 


118.596 


3 
4 


I 5°3'3 


J 37 445 


7 
8 


797.98 


100.138 


7 
8 


1126.7 


118.988 


I 


1511.9 


^37^37 


b 2 


804.25 


100.531 


38 


1134.1 


119.381 


44 


!52o-5 


138.230 


i 


810.54 


100.924 


1 
8 


1141.6 


H9-773 


ft 


1529.2 


138.623 


1 

4 


816.86 


101.316 


1 
4 


1149.1 


120.166 


1 
4 


1537-9 


139-015 


§ 


823.21 


101.709 


3 

8 


1156.6 


120.559 


| 


1546.6 


139.408 


1 

2 


829.58 


102.102 


1 

2 


1 164.2 


120.951 


1 
2 


r 555*3 


139.801 


f 


8 35-97 


102.404 


1 


1171.7 


121.344 


5 

8 


1564.0 


140.194 


3. 
4 


842.3a 


102.887 


3 
4 


1179.3 


121.737 


3. 

4 


1572.8 


140.586 


i 


848.83 


103.280 


1 


1 186.9 


122.129 


7 

8 


1581.6 


140.979 


33 


855.30 


103.673 


39 


1 194.6 


122.522 


45 


1590.4 


141.372 


§ 


861.79 


104.065 


i 


1202.3 


122.915 


i 


1599.3 


141.764 


1 

4 


868.31 


104.458 


1 
4 


1210.0 


123.308 


1 


1608.2 


142.157 


1 


874.85 


104.851 


§ 


1217.7 


123.700 


1 


1 61 7.0 


142.550 


4 


881.41 


105.243 


i 


1225.4 


124.093 


1 

2 


1626.0 


142.942 


f 


888.00 


105.636 


5 

8 


1233.2 


124.486 


i 


1634.9 


J43-335 


3 
4 


894.62 


106.029 


4 


1241.0 


124.878 


3 
4 


1643.9 


143.728 


8 


901.26 


T06.421 


s 


1248.8 


125.271 


8 


1652.0 


144. 121 



193 



TABLE I, 



-AREAS AXD CIRCUMFEREN'CES OF CIRCLES FROM 
i TO ioo {Continued) 



Dia. 


Area 


Circum. 


Dia. 


Area 


1 Circum. 


Dia. 


Area 


1 Circum. 


46 


1 661. 9 


144.513 


52 


2123.7 


163.363 


58 


2642.1 


182.212 


i 


1670.9 


144.906 


i 


2133-9 


163.756 


1 

8 


2653-5 


182.605 


i 


16S0.0 


145.299 


1 
4 


2144.2 


164.143 


1 

4 


2664.9 


182.998 


1 


1689. 1 


145.691 


3 

8 


2154.5 


164.541 


3 

8 


2676.4 


183.390 


i 


1698.2 


146.084 


i 


2164.8 


164.934 


1 
2 


2687.8 


*&3.7%3 


f 


1707.4 


146.477 


5 

8 


2I75-I 


165.326 


f 


2699.3 


184.176 


1 


1716.5 


146.869 


3 

4 


2185.4 


165.719 


3. 

4 


2710.9 


184.569 


7 
5 


I725-7 


147.262 


I 


2195.8 


166. 112 


7. 
8 


2722.4 


184.961 


47 


1734.9 


T47-655 


53 


2206.2 


166.504 


59 


2734.0 


185.354 


I 


1744.2 


148.048 


1 


2216.6 


166.897 


i 


2745.6 


185.747 


i 


1753-5 


148.440 


i 


2227.0 


167.290 


i 


2 757-2 


186.139 


§ 


1762.7 


148.833 


§ 


2237-5 


167.683 


I 


2768.8 


186.532 


i 


1772. 1 


149.226 


1 

9 


2248.0 


168.07:; 


1 
2 


2780.5 


186.925 


i 


1781.4 


149.618 


5 

8 


2258.5 


168.468 


5 

8 


2792.2 


^7-3*1 


l 


1790.8 


150.011 


3 
4 


2269.1 


168.861 


3 
4 


2803.9 


187.710 


1 


1800. 1 


150.404 


7 
8 


2279.6 


169.253 


i 


2815.7 


188.103 


48 


1809.6 


150.796 


54 


2290.2 


169.646 


60 


2827.4 


188.496 


i 


1819.0 


151. 189 


1 

8 


2300.8 


170.039 


i 


2839.2 


188.888 


i 


1828.5 


l5l-5 8 2 


1 
4 


23II-5 


170.431 


i 


2851.O 


189.281 


3. 

8 


18379 


I5I-975 


| 


2322.1 


170.824 


1 


2862.9 


1S9.674 


\ 


iS47-5 


1 5 2 -3 6 7 


i 


2332.8 


171. 217 


0- 


2874.8 


190.066 


I 


1857.0 


152.760 


5 

8 


2343o 


171 .609 


5 

8 


2886.6 


190.459 


3 
4 


1866.5 


J 53-i53 


3 

4 


2 354-3 


172.002 


3 
4 


2898.6 


190.852 


I 


1876.1 


153-544 


1 


2365.0 


172.395 


i 


2910.5 


191.244 


49 


1885.7 


I53-938 


55 


2375-8 


172.788 


61 


2922.5 


191.637 


1 


1895.4 


154-33* 


i 


2386.6 


173.180 


I 


2934o 


192.030 


1 

4 


1905.0 


154.723 


1 
4 


2397-5 


J 73-573 


1 
4 


2946.5 


192.423 


§ 


1914.7 


i55- 116 


1 


2408.3 


173.966 


1 


2958.5 


192.815 


i 


1924.2 


I55-509 


i 


2419.2 


I74-358 


i 


2970.6 


193.208 


i 


1934.2 


I55-904 


f 


2430.1 


i74.75i 


5 

8 


2982.7 


193.601 


i 


1943-9 


156.294 


3 

4 


2441. 1 


I75-I44 


f 


2994.8 


193-993 


7 
8 


J 953-7 


156.687 


7 


2452.0 


I75-536 


I 


3006.9 


194.3S6 


50 


1963-5 


157.080 


56 


2463.0 


I75-929 


62 


3019. 1 


194-779 


1 


1973-3 


I57-472 


i 


2474.0 


176.322 


i 


303I-3 


I95-I7I 


1 
4 


1983.2 


157.865 


i 


2485.0 


176.715 


1 
4 


3°43-5 


I95-564 


1 


I993-I 


158.258 


1 


2496.1 


177.107 


1 


3°55-7 


195-957 


1 


2003.0 


158.650 


1 


2507.2 


177.500 


i 


3068.0 


196.350 


f 


2012.9 


159.043 


5 

8 


2518.3 


I77-893 


1 


3080.3 


196.742 


f 


2022.8 


159.436 


J 


2529-4 


178.285 


1 
4 


3092.6 


197-135 


1 


2032.8 


159.829 


7 
8 


2540.6 


178.678 


i 


3 io 4-9 


197.528 


51 


2042.8 


160.221 


57 


2551.8 


179.071 


63 


3117.2 


197.920 


i 


2052.8 


160.614 


i 


2563.0 


179.463 


i 


3129.6 


198.313 


i 


2062.9 


161.007 


1 
4 


2574.2 


179.856 


1 


3142.0 


198.706 


I 


2073.0 


161.399 


3 

8 


25854 


180.249 


3 

8 


3154.5 


199.098 


4 


2083.1 


161.792 


i 


2596.7 


180.642 


i 


3166.9 


199491 


1 


2093.2 


162.185 


8 


2608.0 


181.034 


1 


3!79-4 


199.8S4 


i 


2103.3 


162.577 


3 
4 


2619.4 


181.427 


1 
4 


3I9I-9 


200.277 


I 


2H3-5 


162.970 


I 


2630.7 


181.820 


i 


3204.4 


200. 66q 



194 



TABLE I.— AREAS AND CIRCUMFERENCES 


> OF 


CIRCLES FROM 




1 TO 100 (C 


ontinued) 






Dia. 


Area 


Circum. 


Dia. 


Area 


Circum. 


Dia. 


Area 


Circum. 


64 


3217.0 


201.062 


70 


3848.5 


219.911 


7°~ 


4536.5 


238.761 


1 

8 


3229.6 


201.455 


1 

8 


3862.2 


220.304 


1 

8 


45514 


239-I54 


i 


3242.2 


201.847 


1 

4 


3876.0 


220.697 


1 

4 


4566.4 


239.546 


1 


3 2 54.8 


202.240 


3 

8 


3889.8 


221.090 


1 


4581.3 


239.939 


4 


3 26 7.5 


202.633 


1 

2 


3903-6 


221.482 


4 


4596.3 


240332 


§ 


3280.1 


203.025 


5 

8 


3917-5 


221.875 


f 


461 1. 4 


240.725 


I 


3292.8 


203.418 


3 
4 


3931-4 


222.268 


3 
4 


4626.4 


241. 117 


i 


33°5- 6 


203.811 


7 
8 


3945-3 


222.660 


7 
8 


4641.5 


241.510 


65 


33i 8 -3 


204.204 


71 


3959-2 


223.053 


77 


4656.6 


241.903 


I 


333 1 - 1 


204.596 


1 


3973-1 


223.446 


1 

8 


4671.8 


242.295 


1 
4 


3343-9 


204.989 


1 

4 


3987.1 


223.838 


1 

4 


4686.9 


242.688 


1 


335 6 -7 


205.382 


3 

8 


4001. 1 


224.231 


3 
5 


4702.1 


243.081 


1 

5 


3369.6 


205.774 


4 


4015.2 


224.624 


1 


47I7.3 


243-473 


1 


3382.4 


206.167 


5 

8 


4029.2 


225.017 


5 

8 


4732.5 


243.866 


f 


3395-3 


206.560 


3 

4 


4043-3 


225.409 


3 
4 


4747-8 


244.259 


1 


3408.2 


206.952 


7 
8 


4057-4 


225.802 


7 
8 


4763.1 


244.652 


66 


3421.2 


207.345 


72 


4071.5 


226.195 


78 


4778.4 


245.044 


1 

8 


3434-3 


207.738 


1 

8 


4085.7 


226.587 


i 


4793-7 


245-437 


1 
4 


3447-2 


208.131 


1 
4 


4099 .8 


226.930 


1 
4 


4809.0 


245.830 


1 


3460.2 


208.523 


3 

8 


4114.0 


227.373 


3 

8 


4824.4 


246.222 


4 


3473-2 


208.916 


4 


4128.2 


227.765 


1 



4839.8 


246.615 


1 


3486.3 


209.309 


5 

8 


4142.5 


228.158 


1 


4855.2 


247.008 


1 


3499-4 


209.701 


3 
4 


4156.8 


228.551 


3 
4 


4870.7 


247.400 


I 


35 I2 -5 


210.094 


i 


4171.1 


228.944 


i 


4886.2 


247-793 


67 


3525-7 


210.487 


13 


4185.4 


229.336 


79 


4901.7 


248.186 


J 


3538.8 


210.879 


1 

8 


4199-7 


229.729 


1 

8 


4917.2 


248.579 


1 


3552.o 


211.272 


1 

4 


4214. 1 


230.122 


i 


4932.7 


248.971 


1 


3565-2 


211.665 


3 

8 


4228.5 


230.514 


3 

8 


4948.3 


249.364 


J 


3578.5 


212.058 


1 

2 


4242.9 


230.907 


1 

2 


4963.9 


249-757 


f 


359L7 


212.450 


5 

8 


42574 


231.300 


5 

8 


4979-5 


250.149 


a 
4 


3605.0 


212.843 


3 
4 


4271.8 


231.692 


3 
4 


4995.2 


250.542 


1 


3618.3 


213.236 


I 


4286.3 


232.085 


7 
8 


5010.9 


25 -935 


68 


3631.7 


213.628 


74 


4300.8 


232.478 


80 


5026.5 


251-327 


1 

8 


3645 -° 


214.021 


1 

8 


43I5-4 


232.871 


i 


5042.3 


251.720 


i 


3658.4 


214.414 


1 
4 


4329.9 


233.263 


1 
• 4 


5058.0 


252.113 


| 


3671.8 


214.806 


| 


4344-5 


233-656 


f 


5073-8 


252.506 


4 


3685.3 


215.199 


1 
2 


4359-2 


234.049 


i 


5089.6 


252.898 


5 

8 


3698.7 


2I5-592 


5 

8 


4373-8 


234.441 


5 

8 


5I054 


253.291 


3 
4 


3712.2 


215.984 


3 

4 


4388.5 


234-334 


3 

4 


5121.2 


253.684 


1 


37257 


216.337 


8 


4403.1 


235.227 


i 


5i37.i 


254.076 


69 


3739-3 


216.770 


75 


4417.9 


235.619 


81 


5 J 53-o 


254.469 


i 


3752.8 


217.163 


i 


4432.6 


236.012 


i 


5168.9 


254.862 


i 


3766.4 


217-555 


1 

4 


4447.4 


236.405 


1 

4 


5184.9 


255.254 


i 


3780.0 


217.948 


§ 


4462.2 


236.798 


3 

8 


5200.8 


255.647 


i 


3793-7 


218.341 


•4 


4477.0 


237.190 


i 


5216.8 


256.040 


f 


3807.3 


218.733 


5 

8 


4491.8 


237.583 


5 

8 


5232.8 


256433 


i 


3821.0 


219.126 


3 
4 


4506.7 


237.976 


3 
4 


5248.9 


256.825 


I 3834.7 


219.519 


7 
8 


4521.5 


238.368 


i 


5264.9 


257.218 



14 



195 



TABLE I.— AREAS AND CIRCUMFERENCES OF CIRCLES FROM 
I TO ioo (Continued) 



Did. 


Area 


82 


5281.O 


1 

8 


5 2 97-l 


1 

4 


53I3-3 


i 
1 
2 

1 

3 
4 


53294 
5345-6 
5361.8 
5378.1 


7 

8 


5394.3 


83 

1 

8 


5410.6 
5426.9 


i 

f 


5443-3 
5459-6 
5476.0 


5 

8 

I 


5492.4 
5508.8 


* 


5525-3 


84 

1 
8 
1 
4 


5541.8 
5558.3 
5574-8 


i 

i 
I 

3_ 
4 

1 


5591-4 
5607.9 

5624.5 
5641.2 

5657.8 


85 

1 
5 


5674.5 
5691.2 


i 


5707.9 


i 


5724-7 


i 
f 


574L5 
5758.3 


4 


5775-1 


8 


579L9 


86 

i 
I 

1 

8 


5808.8 

5825.7 
5842.6 

5859.6 
5876.5 
5893.5 


3_ 
4 


5910.6 


8 


5927.6 


87 
i 

1 
4 

1 
i 

a 
4 


5944-7 
5961.8 

5978.9 
5996.0 
6013.2 
6030.4 
6047.6 
6064.9 



Circum. 


JDia. 


Area 


Circum. 


|Dia. 


Area 


257.611 


88 


6082.1 


276.460 


94 


6939.8 


258.OO3 


1 

8 


6099.4 


276.853 


i 


6958.2 


258.396 


i 


6256.7 


277.846 


i 


6976.7 


258.789 


1 


6134.1 


277.638 


3 

8 


6995.3 


259.181 


J 


61514 


278.031 


i 


7013.8 


259.574 


5 

8 


6168.8 


278.424 


5 

8 


7032.4 


259.967 


i 


6186.2 


278.816 


I 


7051.0 


260.359 


i 


6203.7 


279.209 


I 


7069.6 


260.752 


89 


6221. 1 


279.602 


95 


7088.2 


261.145 


i 


6238.6 


279.994 


i 


7106.9 


261.538 


1 
4 


6256.1 


280.387 


1 
4 


7125.6 


261.930 


f 


62737 


280.780 


1 


7M4.3 


262.323 


i 


6291.2 


281.173 


1 

2 


7163.0 


262.716 


f 


6308.8 


281.565 


5 

8 


7181.8 


263.103 


3 
4 


6326.4 


281.958 


3 

4 


7200.6 


263.501 


7 
8 


6344.1 


282.351 


i 


7219.4 


263.894 


90 


6361.7 


282.743 


96 


7238.2 


264.286 


i 


63794 


283.136 


i 


7 2 57.i 


264.679 


1 

4 


6307.1 


283.529 


1 

4 


7276.0 


265.072 


3 

8 


6414.9 


283.921 


3 

8 


7294.9 


265.465 


1 


6432.6 


284.314 


1 


7313-8 


265.857 


5 

8 


6450.4 


284.707 


I 


7332.8 


266.250 


3 

4 


6468.2 


285.100 


3 
4 


7351-8 


266.643 


i 


6486.O 


285.492 


7 
8 


7370.8 


267.035 


91 


6503.9 


285.885 


97 


7389.8 


267.428 


i 


6521.8 


286.278 


i 


7408.9 


267.821 


1 
4 


65397 


286.670 


i 


7428.0 


268.213 


1 


6557.6 


287.063 


3 

8 


7447-1 


268.606 


i 


6575.5 


287.456 


1 
2 


7466.2 


268.999 


5 
8 


6593.5 


-287.848 


5 

8 


7485.3 


269.392 


3 

4 


66 1 1. 5 


288.241 


3 

4 


7504.5 


269.784 


i 


6629.6 


288.634 


8 


75237 


270.177 


92 


6647.6 


289.027 


98 


7543.0 


270.570 


i 


666s. 7 


289.419 


1 

8 


7562.2 


270.962 


1 
4 


6683.8 


289.812 


1 
4 


7581.5 


27L355 


3 
s 


6701.9 


290.205 


3 

8 


7600.8 


271.748 


1 


6720.1 


290.597 


1 
2" 


762O. T 


272.140 


5 

8 


6738.2 


290.990 


5 

8 


7639.5 


272.533 


3 

4 


6756.4 


291.383 


3 
4 


7658.9 


272.926 


1 


6774.7 


291.775 


I 
8 


7678.3 


2 73-3 I 9 


93 


6792.9 


292.168 


99 


7697.7 


273.711 


i 


681 1. 2 


292.561 


i 


77I7.I 


274.104 


i 


6829.5 


292.954 


i 


7736.6 


274.497 


3 

8 


6847.8 


293.346 


3 

8 


7756.1 


274.889 


1 
2 


6866.1 


293-739 


I 


7775-6 


275.282 


8 


6884.5 


294.132 


5 
8 


7795-2 


275.675 


3 
4 


6902.9 


294.524 


3 
4 


7814.8 


276.067 


7 
8 


6921.3 


294.917 


7 
8 


78344 



196 



TABLE II.— DECIMAL EQUIVALENTS OF FRACTIONS OF AN 

INCH. (ADVANCING BY 8THS, 16THS, 32NDS AND 

64THS.) 



8ths 


3- 


>nds 


64ths 


64ths 


I = - T2 5 


1 _ 

.32 — 


•°3 I2 5 


A = .015625 


If = .515625 


i = -250 


3 

32 — 


•°9375 


A = .046875 


ff = .546875 


1 = -375 


32 = 


.15625 


A = .°7 8 i25 


H - -578125 


1 = .500 


3 2 = 


.21875 


A = - io 9375 


If = .609375 


f - .625 


9 _ 
3 2 — 


.28125 


A = -140625 


tt = .640625 


i = -75o 


11 

3 2 — 


•34375 


*i - -171875 


H = .671875 


i - .875 


13 

"3"2 — 


.40625 


« = -203125 


It = .7°3 I2 5 




15 

32 — 


46875 


a = .234375 


H = .734375 


i6ths. 










A = -0625 


17 _ 
3 2 — 


03125 


a = .265625 


f| = .765625 


A - .1875 


« = 


•59375 


H = .296875 


« = .796875 


A = -3I25 


2 1 — 
32 — 


.65625 


*i - -328125 


II - .828125 


A = -4375 


2 3 

32 — 


•71875 


If = .359375 


H = .859375 


A = 0625 


25 _ 

32 — 


.78125 


If = .390625 


II = .890625 


tt - -6875 


2 7 

32 — 


.84375 


U = .421875 


If = .921875 


if - .8125 


29 

3 2 — 


.90625 


II = -453 I2 5 


ft = .953125 


x# = -9375 


31 

3 2 — 


.96875 


tt = 484375 


If = .984375 



197 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 




TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 {Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


41 


1681 


68921 


6.4031 


3.4482 


128.81 


1320.25 


42 


1764 


74088 


6.4807 


3.4760 


I3I-95 


138544 


43 


1849 


795°7 


6-5574 


3-5°34 


I35.09 


1452.20 


44 


1936 


85184 


6.6332 


3-53°3 


138.23 


1520.53 


45 


2025 


91125 


6.7082 


3o569 


141.37 


I590.43 


46 


2116 


9733 6 


6.7823 


3-5830 


144-5 l 


1661.90 


47 


2209 


103823 


6.8557 


3.6088 


147.65 


1734-94 


48 


2304 


1 10592 


6.9282 


3-6342 


150.80 


1809.56 


49 


2401 


1 1 7649 


7.0000 


3.6-93 


153.94 


1885.74 


5° 


2500 


125000 


7.0711 


3.6840 


157.08 


1963.50 


51 


2601 


132651 


7.1414 


3.7084 


160.22 


2042.82 


5 2 


2704 


140608 


7.2111 


3-7325 


163.36 


2123.72 


53 


280Q " 


148877 


7.2801 


3-7563 


166.50 


2206.18 


54 


2916 


157464 


7-3485 


3-7798 


169.65 


2290.22 


55 


3° 2 5 


166375 


7.4162 


3.8030 


172.79 


2375.83 


56 


3 T 3 6 


175616 


74833 


3-8259 


175.93 


2463.OI 


57 


3 2 49 


185193 


7-5498 


3.8485 


179.07 


2551-76 


58 


3364 


195112 


7.6158 


3.8709 


182.21 


2642.08 


59 


348i 


205379 


7.6811 


3-893° 


185.35 


2733.97 


60 


3600 


2 1 6000 


7.7460 


3-9!49 


188.50 


2827.43 


61 


3721 


226981 


7.8102 


3-9365 


191.64 


2922.47 


62 


3 8 44 


238328 


7.8740 


3-9579 


194.78 


3019.07 


63 


3969 


250047 


7-9373 


3-9791 


197.92 


3II7.25 


64 


4096 


262144 


8.0000 


4.0000 


201.06 


3216.99 


65 


4225 


274625 


8.0623 


4.0207 


204.20 


33^-3* 


66 


435 6 


287496 


8.1240 


4.0412 


207.35 


3421.19 


67 


4489 


300763 


8.1854 


4.0615 


210.49 


3525-65 


68 


4624 


3 J 443 2 


8.2462 


4.0817 


213.63 


3631.68 


69 


4761 


328509 


8.3066 


4.1016 


216.77 


3739.28 


70 


4900 


343000 


8.3666 


4-1213 


219.91 


384845 


7i 


5°4i 


3579" 


8.4261 


4.1408 


223.05 


3059. T 9 


72 


5184 


373248 


8.4853 


4.1602 


226.19 


4071.50 


73 


53 2 9 


389017 


8.5440 


4.1793 


229.34 


4185.39 


74 


5476 


405224 


8.6023 


4.1983 


232.48 


4300.84 


75 


5625 


421875 


8.6603 


4.2172 


235.62 


4417.86 


76 


5776 


438976 


8.7178 


4.2358 


238.76 


453646 


77 


5929 


456533 


8.775o 


4-2543 


241.90 


4656.63 


78 


6084 


474552 


8.8318 


4.2727 


245.04 


4778.36 


79 


6241 


493°39 


8.8882 


4.2908 


248.19 


4001.67 


80 


6400 


512000 


8.9443 


4.3089 


251.33 


5o 2 6.55 



199 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERE.XCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 













ClR^T T 


No. 


Square 


Cube 


Sq. Root 


Cube Root 






Circum. 


Area 


8i 


6561 


53 T 44i 


9 .OOOO 


4-3267 


254-47 


5 J 53-oo 


82 


6724 


551368 


9-0554 


4-3445 


257.61 


5281.02 


83 


6889 


571787 


9.1104 


4.3621 


260.75 


5410.61 


84 


7056 


592704 


9.1652 


4-3795 


263.89 


5541-77 


85 


7225 


614125 


9-2195 


4.3968 


267.04 


5674.50 


86 


7396 


636056 


9.2736 


4.4140 


270.18 


5808.80 


87 


7569 


658503 


9-3274 


4.4310 


273-32 


5944-68 


88 


7744 


681472 


9.3808 


4.4480 


276.46 


6082.12 


89 


7921 


704969 


9.4340 


4.4647 


279.60 


6221. 14 


90 


8100 


729000 


9.4868 


44S14 


282.74 


6361.73 


9i 


8281 


753571 


9-5394 


4-4979 


285.88 


6503.88 


92 


8464 


778688 


9-5917 


4-5*44 


289.03 


6647.61 


93 


8649 


8o4357 


9-6437 


4.5307 


292.17 


6792.91 


94 


8836 


830584 


9.6954 


4.5468 


295.3I 


6939.78 


95 


9025 


857375 


9.7468 


4.5629 


298.45 


7088.22 


96 


Q2l6 


884736 


9.7980 


4.57 8 9 


301.59 


7238.23 


97 


9409 


912673 


9.8489 


4-5947 


304.73 


7389.81 


98 


9604 


941192 


9.8995 


4.6104 


307.88 


7542.96 


99 


9801 


970299 


9.9499 


4.6261 


311.02 


7697.69 


100 


I OOOO 


I 000000 


10.0000 


4.6416 


314.16 


7853.98 


IOI 


I020I 


1030301 


10.0499 


4.6570 


3*7-30 


8011.85 


102 


IO404 


1061208 


10.0995 


4-6723 


320.44 


8171.28 


103 


I0609 


1092727 


10.1489 


4.6875 


3 2 3-58 


8332.29 


104 


I0816 


1 1 24864 


10.1980 


4.7027 


3 26 -73 


8494.87 


105 


IIO25 


II57625 


10.2470 


4.7177 


329.87 


8659.OI 


106 


II236 


1191016 


10.2956 


4.7326 


333-0 1 


8824.73 


107 


1 1 449 


1225043 


10.3441 


4-7475 


336.15 


8992.02 


108 


1 1 664 


1259712 


10.3923 


4.7622 


339-29 


9160.88 


109 


11881 


1295029 


10.4403 


4.7769 


342.43 


933I-32 


no 


12100 


1331000 


10.4881 


4.7914 


345-58 


9503-32 


III 


12321 


1367631 


io.5357 


4.8059 


348.72 


9676.89 


112 


12544 


1404928 


10.5830 


4.8203 


351.86 


9852.03 


113 


12769 


1442897 


10.6301 


4.8346 


35 5 -oo 


IOO28.7 


TI4 


12996 


I 48 1 544 


10.6771 


4.8488 


358.14 


IO207.O 


"5 


13225 


1520875 


10.7238 


4.8629 


361.28 


IO386.9 


Il6 


I345 6 


1560896 


10.7703 


4.8770 


364.42 


IO568.3 


117 


13689 


1601613 


10.8167 


4.8910 


367o7 


I075I-3 


Il8 


13924 


1643032 


10.8628 


4.9049 


37o.7i 


10935-9 


119 


14161 


1 685 1 59 


10.0087 


4.Q187 


373-85 


III22.0 


I20 


14400 


1728000 


10.9545 


4.9324 


376.99 


I I309.7 



200 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENXES AXD CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 













ClRC'i -V. 


No. 


Square 


Cube 


Sq. Root 


Cube Root 






Circum. 


Area 


121 


14641 


1771561 


1 1 .OOOO 


4.9461 


380.13 


1 1 499.O 


122 


14884 


1815848 


11.0454 


4-9597 


383.27 


1 1 689 .9 


123 


15129 


1860867 


II.0905 


4.9732 


386.42 


11882.3 


124 


15376 


1906624 


H-I355 


4.9866 


38906 


12076.3 


125 


15625 


i953 I2 5 


11. 1803 


5 .OOOO 


392.70 


12271.8 


126 


15876 


2000376 


11.2250 


5-°i33 


395.84 


12469.O 


127 


16129 


2048383 


11.2694 


5.0265 


398.98 


12667.7 


128 


16384 


2097152 


"•3137 


5-°397 


402.12 


12868.O 


129 


1 664 1 


2146689 


11.3578 


5.0528 


405.27 


13069.8 


13° 


16900 


2197000 


1 1. 4018 


5.0658 


408.41 


13273.2 


131 


17161 


2248091 


n-4455 


5.0788 


411-55 


13478.2 


132 


17424 


2299968 


1 1. 489 1 


5.0916 


414.69 


13684.8 


133 


17689 


235 26 37 


11.5326 


5.1045 


417.83 


13892.9 


134 


I795 6 


2406104 


11.5758 


5-1172 


420.97 


14102.6 


135 


18225 


2460375 


11. 6190 


5.1299 


424.12 


I43I3.9 


136 


18496 


2515456 


1 1. 6619 


5.1426 


427.26 


14526.7 


137 


18769 


2571353 


11.7047 


5-I55 1 


430.40 


14741.I 


138 


19044 


2628072 


11-7473 


5.1676 


433-54 


I4957-I 


139 


19321 


2685619 


11.7898 


5.1801 


436.68 


15*74-7 


140 


19600 


2744000 


11.8322 


5- I 9 2 5 


439.82 


*5393& 


141 


19881 


2803221 


11.8743 


5.2048 


442.96 


15614-5 


142 


20164 


2863288 


1 1. 9 1 64 


5.2171 


446.II 


15836.8 


143 


20449 


2924207 


n.9583 


5.2293 


449.2 5 


16060.6 


144 


20736 


2985984 


12.0000 


5.2415 


452.39 


16286.0 


J 45 


21025 


3048625 


12.0416 


5-2536 


455-53 


1 6513.0 


146 


21316 


3112136 


12.0830 


5.2656 


458.67 


16741.5 


147 


21609 


3176523 


12.1244 


5.2776 


461.81 


16971.7 


148 


21904 


3241792 


12.1655 


5.2896 


464.96 


17203.4 


149 


22201 


3307949 


12.2066 


5.3015 


468.IO 


17436.6 


15° 


22500 


3375°°° 


12.2474 


5.3133 


471.24 


I767I-5 


151 


22801 


344295 1 


12.2882 


5.3251 


474.38 


17907.9 


152 


23104 


35 1 1808 


12.3288 


5.3368 


477.52 


18145.8 


153 


23409 


358i577 


12.3693 


5.3485 


480.66 


18385.4 


154 


23716 


3652264 


12.4097 


5.3601 


483.81 


18626.5 


155 


24025 


372387=: 


12.4499 


5.3717 


486.95 


18869.2 


156 


24336 


3796416 


12.4900 


5.3832 


490.09 


191134 


157 


24649 


3869893 


12.5300 


5-3947 


493-23 


19359.3 


158 


24964 


3944312 


12.5698 


5.4061 


496.37 


19606.7 


159 


25281 


4019679 


I2.0OQ5 


5.4175 


499 -5 1 


19855.7 


160 


25600 


4096000 


I2.649I 


54288 


502.65 


20106.2 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


161 


25921 


4173281 


12.6886 


5.4401 


505.80 


20358.3 


162 


26244 


4251528 


12.7279 


5-45 J 4 


508.94 


20612.0 


163 


26^69 


433°747 


12.7671 


5.4626 


512 .08 


20867.2 


164 


26896 


4410944 


12.8062 


5-4737 


515.22 


21124.1 


165 


27225 


4492125 


I2.S452 


5.4848 


518.36 


213S2.5 


166 


27556 


4574296 


12.8841 


5-4959 


521.50 


21642.4 


167 


27889 


4657463 


12.9228 


5-5069 


524.65 


21904.0 


168 


2G224 


4741632 


12.9615 


5.5178 


527-79 


22167. 1 


169 


28561 


4826809 


1 3 .0000 


5.5288 


530.93 


22431.8 


170 


28900 


4913000 


13.0384 


5-5397 


534.07 


22698.O 


171 


29241 


500021 T 


13.0767 


5-5505 


537-21 


22965.8 


172 


29584 


5088448 


13.1149 


5-56i3 


540.35 


23235.2 


173 


29929 


5!777 J 7 


*3-*S*9 


5-5721 


543-5° 


23506.2 


174 


30276 


5268024 


13.1909 


5.5828 


546.64 


23778.7 


175 


30625 


5359375 


13.2288 


5-5934 


549-78 


24052.8 


176 


30976 


5451776 


13.2665 


5.6041 


552.92 


24328.5 


177 


3*3 2 9 


5545233 


13-3041 


5.6i47 


556.o6 


24605.7 


178 


31684 


5639752 


I3-34I7 


5.6252 


559-20 


24884.6 


179 


32041 


5735339 


^3-3791 


5.6357 


562.35 


25164.9 


180 


32400 


5832000 


13.4164 


5.6462 


56549 


25446.9 


181 


32761 


5929741 


I3-4536 


5.6567 


568.63 


25730.4 


182 


33 I2 4 


6028568 


13.4907 


5.6671 


57L77 


26015.5 


183 


33489 


6128487 


I3-5277 


5.6774 


574.91 


26302.2 


184 


33856 


6229504 


I3-5647 


5.6877 


578.05 


26^90.4 


185 


34225 


6331625 


13.6015 


5.6980 


581.19 


26880.3 


186 


34596 


6434856 


13.6382 


5.7083 


584.34 


27171.6 


187 


34969 


6539203 


13.6748 


57185 


587.48 


27464.6 


188 


35344 


6644672 


i3-7ii3 


5.7287 


590.62 


27759.1 


189 


35721 


6751269 


13-7477 


57388 


59376 


28055.2 


190 


36100 


6859000 


15.7840 


5-7489 


596.90 


28352.9 


191 


36481 


6967871 


13.8203 


5.7590 


600.04 


28652.I 


192 


36864 


7077888 


13.8564 


5.7690 


603.19 


28952.9 


193 


37 2 49 


7189057 


13.8924 


5.7790 


606.33 


29255.3 


194 


37636 


73 OI 384 


13.9284 


5.7890 


609.47 


29559.2 


195 


38025 


7414875 


13.9642 


57989 


612.61 


29864.8 


196 


38416 


7529536 


14.0000 


5.8088 


6i5.75 


30171.9 


197 


38809 


7645373 


14.0357 


5.8186 


618.89 


30480.5 


198 


39204 


7762392 


14.0712 


5.8285 


622.04 


30790.7 


199 


39601 


7880599 


14.1067 


5-8383 


625.18 


31102.6 


200 


40000 


8000000 


14.1421 


5.8480 


628.32 


3I4I5.9 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 













ClI?^T TT 


No. 


Square 


Cube 


Sq. Root 


Cube Root 






Circum. 


Area 


201 


40401 


8120601 


14.1774 


5.8578 


631.46 


31730.9 


202 


40804 


8242408 


14.2127 


5.8675 


634.60 


320474 


203 


41209 


8365427 


14.2478 


5-877I 


637.74 


32365.5 


204 


41616 


8489664 


14.2829 


5.8868 


640.89 


32685.1 


205 


42025 


8615125 


14.3178 


5.8964 


644.03 


33006.4 


206 


42436 


8741816 


I4.3527 


5-9059 


647.17 


33329.2 


207 


42849 


8869743 


14.3875 


5-9*55 


650.31 


33653-5 


208 


43 2 64 


8998912 


14.4222 


5-9250 


653-45 


33979-5 


209 


43681 


9129329 


14.4568 


5-9345 


656.59 


34307.0 


210 


44100 


9261000 


14.4914 


5-9439 


659-73 


34636.1 


211 


44521 


939393 1 


14.5258 


5-9533 


662.88 


34966.7 


212 


44944 


9528128 


14.5602 


5.9627 


666.02 


35298.9 


213 


45369 


9663597 


14.5945 


5.9721 


669.16 


35632.7 


214 


45796 


9800344 


14.6287 


5.9814 


672.30 


35968.1 


215 


46225 


9938375 


14.6629 


5.9907 


675-44 


36305.O 


216 


46656 


10077696 


14.6969 


6.0000 


678.58 


36643.5 


217 


47089 


10218313 


14.7309 


6.0092 


681.73 


36983.6 


218 


47524 


10360232 


14.7648 


6.0185 


684.87 


37325-3 


219 


47961 


io 5°3459 


14.7986 


6.0277 


688.01 


37668.5 


220 


48400 


10648000 


14.8324 


6.0368 


691.15 


38013.3 


221 


48841 


10793861 


14.8661 


6.0459 


694.29 


38359.6 


222 


49284 


1 094 1 048 


14.8997 


6.0550 


69743 


38707.6 


223 


49729 


11089567 


14.9332 


6.0641 


700.58 


39057.1 


224 


50176 


11239424 


14.9666 


6.0732 


703.72 


39408.I 


225 


50625 


1 1390625 


15.OOOO 


6.0822 


706.86 


39760.8 


226 


51076 


11543*76 


*5-°333 


6.0912 


710.OO 


40115.O 


227 


5^29 


1 1 697083 


15.0665 


6.1002 


7 I 3-*4 


40470.8 


228 


5!9 8 4 


11852352 


15.0997 


6.1091 


716.28 


40828.I 


229 


52441 


12008989 


I 5-i3 2 7 


6.1180 


719.42 


41187.1 


230 


52900 


1 2 167000 


15.1658 


6.1269 


722.57 


41547.6 


231 


5336i 


12326391 


15.1987 


6.1358 


725-7I 


41909.6 


232 


53824 


12487168 


i5- 2 3 I 5 


6.1446 


728.85 


42273.3 


233 


54289 


12649337 


15.2643 


6.1534 


731-99 


42638.5 


234 


54756 


12812904 


15.2971 


6.1622 


735-13 


43005.3 


235 


55225 


12977875 


I5-3297 


6.1710 


738.27 


43373-6 


236 


55696 


13144256 


15-3623 


6.1797 


741.42 


43743-5 


237 


56169 


I 33 I2 o53 


I5-3948 


6.1885 


744.56 


44115.O 


238 


56644 


13481272 


15.4272 


6.1972 


747.70 


44488.I 


239 


57i2i 


13651919 


I5-4596 


6.2058 


750.84 


44862.7 


240 


57600 


13824000 


15.4919 


6.2145 


753-98 


45238.9 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 {Continued) 



No. 



241 

242 

243 
244 

245 
246 

247 
248 

249 
250 

251 
252 

253 
254 
255 

256 

257 
258 

259 
260 

261 

262 
263 
264 
265 

266 
267 
268 
269 

270 

271 
272 

273 
274 
275 

276 

277 
278 

279 
280 



Square 



58564 
59049 
59536 
60025 

60516 
61009 
61504 
62001 
62500 

63001 

635 4 
64009 
64516 
65025 

65536 
66049 
66564 
67081 
67600 

68l 2 1 
68644 
69169 
69696 

70225 

70756 
71289 
71824 
72361 
729OO 

73441 

73984 
745 2 9 
75076 
75625 

76176 
76729 
77284 
7784I 
784OO 



Cube 



3997521 
4172488 
4348907 
4526784 
4706125 

4886936 
5069223 
5252992 

543 82 49 
5625000 

5813251 
6003008 
6194277 
6387064 
6581375 

6777216 

6974593 
7I735I2 

7373979 
7576000 

777958i 
7984728 
8191447 

8399744 
8609625 

8821096 
9034163 

9248832 
9465109 
9683000 

19902511 
20123648 
20346417 
20570824 
20796875 

21024576 

21253933 
21484952 
2 1 717639 
21952000 



Sq. Root 


Cube Root 


15.5242 


6.2231 


I 5-55 6 3 


6.2317 


15-5885 


6.2403 


15.6205 


6.2488 


15.6525 


6.2573 


15.6844 


6.2658 


15.7162 


6.2743 


15.7480 


6.2828 


15.7797 


6.2912 


15,8114 


6.2996 


15.8430 


6.3080 


15.8745 


6.3164 


15.9060 


6.3247 


15.9374 


^•333° 


15.9687 


6.3413 


16.0000 


6.3496 


16.0312 


6.3579 


16.0624 


6.3661 


16.0935 


6-3743 


16.1245 


6.3825 


16.1555 


6.3907 


16.1864 


6.3988 


16.2173 


6.4070 


16.2481 


6.4151 


16.2788 


6.4232 


16.3095 


6.4312 


16.3401 


6.4393 


16.3707 


6.4473 


16.4012 


6-4553 


16.4317 


6.4633 


16.4621 


6.47*3 


16.4924 


6.4792 


16.5227 


6.4872 


16.5529 


6.4951 


16.5831 


6.5030 


16.6132 


6.5108 


16.6433 


6.5187 


16.6733 


6.5265 


16.7033 


6-5343 


16.7332 


6.5421 



Circle 



Circum. I 



Area 



757-12 
760.27 
76341 
766.55 
769.69 

772.83 

775-97 
779.12 
782.26 
785.40 

788.54 
791.68 
794.82 
797.96 
801. 11 

804.25 

807.39 
810.53 
813.67 
816.81 

819.96 
823.10 
826.24 
829.38 
832.52 

835.66 
838.81 
841.95 
845.09 
848.23 

851.37 
854.51 

857.66 
860.80 
863.94 

867.08 
870.22 

873-36 
876.50 
879.65 



45616.7 
45996.1 
46377.0 

46759.5 
47M3.5 

-47529.2 
47916.4 
48305.1 
48695.5 
49087.4 

49480.9 

49875.9 
50272.6 
50670.7 
51070.5 

5I47L9 
51874.8 
52279.2 
52685.3 
53092.9 

53502.1 
53912.9 
54325-2 
54739-1 
55J54.6 

55571.6 
55990.3 
56410.4 
56832.2 

57255.5 

57680.4 
58106.9 

58534.9 
58964.6 

59395-7 

59828.5 
60262.8 
60698.7 
61 136.2 
6i575. 2 



204 



.TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF XOS. 

FROM i TO 520 (Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


281 


78961 


22188041 


16.7631 


6.5499 


882.79 


62015.8 


282 


795 2 4 


22425768 


16.7929 


6-5577 


885.93 


62458.0 


283 


80089 


22665187 


16.8226 


6.5654 


889.07 


62901.8 


284 


80656 


22906304 


16.8523 


6.5731 


892.21 


63347.1 


285 


81225 


23149125 


16.8819 


6.5808 


895-35 


63794.O 


286 


81796 


23393656 


16.9115 


6.5885 


898.50 


64242.4 


287 


82369 


23639903 


16.9411 


6.5962 


901.64 


64692.5 


288 


82944 


23887872 


16.9706 


6.6039 


904.78 


65144. 1 


289 


83521 


24137569 


17.0000 


6.6115 


907.92 


65597.2 


290 


84100 


24389000 


17.0294 


6.6191 


911.06 


66052.O 


291 


84681 


24642171 


17.0587 


6.6267 


914.20 


66508.3 


292 


85264 


24897088 


17.0880 


6.6343 


9^7-35 


66966.2 


293 


85849 


25153757 


17.1172 


6.6419 


920.49 


67425.6 


294 


86436 


25412184 


17.1464 


6.6494 


923.63 


67886.7 


295 


87025 


25672375 


17.1756 


6.6569 


926.77 


68349-3 


296 


87616 


25934336 


17.2047 


6.6644 


929.91 


68813.5 


297 


88209 


26198073 


i7- 2 337 


6.6719 


933-05 


69279.2 


298 


88804 


26463592 


17.2627 


6.6794 


936.19 


69746.5 


299 


89401 


26730899 


17.2916 


6.6869 


939-34 


70215.4 


300 


90000 


27000000 


17-3205 


6.6943 


942.48 


70685.8 


301 


90601 


27270901 


17-3494 


6.7018 


945.62 


7II57-9 


302 


91204 


27543608 


17.3781 


6.7092 


948.76 


71631.5 


3°3 


91809 


27818127 


17.4069 


6.7166 


951.90 


72106.6 


3°4 


92416 


28094464 


I7-4356 


6.7240 


955-°4 


725834 


3°5 


93° 2 5 


28372625 


17.4642 


6.73*3 


958.19 


73061.7 


306 


93636 


28652616 


17.4929 


6.7387 


•961.33 


7354L5 


307 


94249 


28934443 


17-5214 


6.7460 


964.47 


74023.O 


308 


94864 


29218112 


17-5499 


6.7533 


967.61 


74506.O 


3°9 


9548i 


29503629 


I7-5784 


6.7606 


97o.75 


74990.6 


310 


96100 


29791000 


17.6068 


6.7679 


973-89 


75476.8 


3 11 


96721 


30080231 


17-6352 


6.7752 


977.04 


75964.5 


312 


97344 


3 37I328 


17.6635 


6.7824 


980.18 


76453.8 


3*3 


97969 


30664297 


17.6918 


6.7897 


983-32 


769447 


3M 


98596 


30959144 


17.7200 


6.7969 


986.46 


77437-1 


3^5 


99225 


31255875 


17.7482 


6.8041 


989.60 


77931. 1 


316 


99856 


31554496 


17.7764 


6.8113 


992.74 


78426.7 


3 X 7 


100489 


31855013 


17.8045 


6.8185 


995.88 


78923.9 


318 


101124 


32157432 


17.8^26 


6.8256 


999.03 


79422.6 


3 X 9 


101761 


32461759 


17.8606 


6.8328 


1002.20 


79922.9 


320 


102400 


32768000 


17.8885 


6.8399 


1005.30 


80424.8 



205 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


321 


IO3041 


33076161 


17.9165 


6.8470 


IO08.5 


80928.2 


322 


103684 


33386248 


17.9444 


6.8541 


IOU.6 


81433.2 


3 2 3 


IO4329 


33698267 


17.9722 


6.8612 


IOI4.7 


81939.8 


3 2 4 


104976 


34012224 


18.0000 


6.8683 


1017.9 


82448.O 


3 2 S 


105625 


3432S125 


18.0278 


6.8753 


1 02I.0 


829577 


326 


106276 


34645976 


18.0555 


6.8824 


1024.2 


83469.O 


3 2 7 


106929 


349657 8 3 


18.0831 


6.8894 


1027.3 


83981.8 


328 


107584 


35287552 


18.1108 


6.8964 m 


IO30.4 


84496.3 


3 2 9 


108241 


35611289 


18.1384 


6.9034 ' 


1033.6 


85012.3 


33° 


108900 


35937000 


18.1659 


6.9104 


1036.7 


85529.9 


33 1 


109561 


36264691 


18.1934 


6.9174 


1039.9 


86049.O 


33 2 


IIO224 


36594368 


18.2209 


6.9244 


IO43.O 


86569.7 


333 


I 10889 


36926037 


18.2483 


6.9313 


1046.2 


87092.O 


334 


III556 


37259704 


18.2757 


6.9382 


1049.3 


87615.9 


335 


112225 


37595375 


18.3030 


6.9451 


1052.4 


88141.3 


336 


112896 


37933 56 


18.3303 


6.9521 


I055.6 


88668.3 


337 


II3569 


38272753 


18.3576 


6.9^89 


IO58.7 


89196.9 


33^ 


I 14244 


38614472 


18.3848 


6.9658 


1 06 1. 9 


89727.O 


339 


114921 


38958219 


18.4120 


6.9727 


106^.0 


90258.7 


340 


I I 5600 


39304000 


18.4391 


6.9795 


1068. 1 


90792.O 


34i 


116281 


39651821 


18.4662 


6.9864 


1071.3 


91326.9 


342 


1 1 6964 


40001688 


18.4932 


6.9932 


1074.4 


91863.3 


343 


1 1 7649 


40353 6o 7 


18.5203 


7 .0000 


1077.6 


92401.3 


344 


118336 


40707584 


18.5472 


7.0068 


1080.7 


92940.9 


345 


II9025 


41063625 


18.5742 


7.0136 


1083.8 


93482.O 


346 


119716 


41421736 


18.6011 


7.0203 


I087.O 


940247 


347 


120409 


41781923 


18.6279 


7.0271 


1090. 1 


94569.0 


348 


121 IO4 


42144192 


18.6S48 


7.0338 


I093.3 


95114.9 


349 


I2l8oi 


42508549 


18.6815 


7.0406 


1096.4 


95662.3 


35o 


I225OO 


42875000 


18.7083 


7.0473 


1099.6 


9621 1.3 


35i 


I232OI 


43243551 


I8.7350 


7.0540 


IIO2.7 


96761.8 


352 


I23904 


43614208 


18.7617 


7.0607 


1105,8 


97314.O 


353 


I24609 


43986977 


18.7883 


7.0674 


1 109.0 


97867.7 


354 


I25316 


44361864 


18.8149 


7.0740 


III2.I 


98423.O 


355 


I20025 


44738875 


18.8414 


7.0807 


III5.3 


98979.8 


356 


I26736 


45118016 


18.8680 


7-0873 


II18.4 


99538.2 


357 


I27449 


45499293 


18.8944 


7.0940 


II2I.5 


IOOO98 


358 


128164 


45882712 


18.9209 


7.1006 


II24.7 


IO0660 


359 


I2888l 


46268279 


18.9473 


7.1072 


1 127.8 


IOI223 


360 


I2960O 


46656000 


18.9737 


7.1138 


II3I.O 


101788 



206 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 
CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 
FROM i TO 520 (Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


361 


130321 


47045881 


19.OOOO 


7.1204 


II34.I 


IO2354 


362 


131044 


47437928 


19.0263 


7.1269 


H37-3 


IO2922 


3 6 3 


131769 


47832147 


19.0^26 


7-1335 


1 140.4 


IO3491 


364 


132496 


48228544 


19.0788 


7.1400 


II43.5 


IO4062 


365 


133225 


48627125 


19.1050 


7.1466 


1146.7 


IO4635 


366 


133956 


49027896 


19.1311 


7-I53I 


1149.8 


IO5209 


3 6 7 


134689 


49430863 


19.1572 


7-I596 


H53-0 


105785 


368 


135424 


49836032 


19-1833 


7.1661 


1156.1 


106362 


369 


136161 


50243409 


19.2094 


7.1726 


1159.2 


106941 


37° 


136900 


50653000 


I9-2354 


7.1791 


1 1 62 .4 


IO7521 


37i 


137641 


51064811 


19.2614 


7.1855 


H65.5 


108103 


372 


138384 


51478848 


19.2873 


7.1920 


1168.7 


108687 


373 


139129 


51895117 


19.3132 


7.1984 


1171.8 


IO9272 


374 


139876 


52313624 


19-3391 


7.2048 


I175.O 


IO9858 


375 


140625 


52734375 


19.3649 


7.2112 


II78.I 


I IO447 


376 


I4I376 


53157376 


19.3907 


7.2177 


Il8l.2 


IIIO36 


377 


142129 


53582633 


19.4165 


7.2240 


1184.4 


IH628 


378 


142884 


54010152 


19.4422 


7.2304 


1187.5 


II222I 


379 


143641 


54439939 


19.4679 


7.2368 


H90.7 


II28I5 


380 


144400 


54872000 


19.4936 


7.2432 


1193.8 


II34II 


381 


145161 


553 634i 


19.5192 


7.2495 


II96.9 


I I 4OO9 


382 


145924 


55742968 


19.5448 


7.2558 


1200. 1 


I I4608 


3^3 


146689 


56181887 


19.5704 


7.2622 


1203.2 


II5209 


384 


147456 


56623104 


19-5959 


7.2685 


1206.4 


I I 581 2 


385 


148225 


57066625 


19.6214 


7.2 748 


1209.5 


H6416 


386 


148996 


57512456 


19.6469 


7.2811 


1212.7 


II702I 


3^7 


149769 


57960603 


19.6723 


7.2874 


1215.8 


1 1 7628 


388 


I5°544 


58411072 


19.6977 


7.2936 


1218.9 


H8237 


389 


1513 21 


58863869 


19.7231 


7.2999 


1222. 1 


I 18847 


390 


152100 


59319000 


19.7484 


7.3061 


1225.2 


I 19459 


39i 


152881 


59776471 


19-7737 


7.3124 


1228.4 


120072 


392 


153664 


60236288 


19.7990 


7.3186 


I23L5 


I20687 


393 


154449 


60698457 


19.8242 


7.3248 


I234.6 


I21304 


394 


155236 


61162984 


19.8494 


7-33 10 


I237.8 


I2I922 


395 


156025 


61629875 


19.8746 


7-337 2 


I24O.9 


122542 


396 


156816 


62099136 


19.8997 


7-3434 


I244.I 


I23163 


397 


157609 


62570773 


19.9249 


7.3496 


1247.2 


I23786 


398 


158404 


63044792 


19.9499 


7.3558 


I25O.4 


I244IO 


399 


159201 


63521199 


19.9750 


7.3619 


I253v5 


I25036 


400 


1 60000 


64000000 


20.0000 


7.3684 


I256.6 


I25664 



207 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 
CIRCUMFERENCES, AND CIRCULAR AREAS OF NOS. 
FROM i TO 520 (Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


401 


I 6080 I 


64481201 


20.0250 


7-3742 


1259.8 


126293 


402 


161604 


64964808 


20.0499 


7-38o3 


1262.9 


126923 


403 


162409 


65450827 


20.0749 


7.3864 


1266. 1 


127556 


404 


163216 


65939264 


20.0998 


7.392 5 


1269.2 


128190 


405 


164025 


66430125 


20.1246 


7.3986 


1272.3 


128825 


406 


164836 


66923416 


20.1494 


74047 


1275.5 


129462 


407 


165649 


67419143 


20.1742 


7.4108 


1278.6 


130100 


408 


166464 


67917312 


20.1990 


7.4169 


1281.8 


130741 


409 


167281 


68417929 


20.2237 


7.4229 


1284.9 


13*382 


410 


168100 


68921000 


20.2485 


7.4290 


1288. 1 


132025 


411 


I 6892 I 


69426531 


20.2731 


7.4350 


1 291.2 


132670 


412 


169744 


69934528 


20.2978 


7.4410 


1294.3 


I333I7 


413 


170569 


70444997 


20.3224 


7.4470 


l297o 


133965 


414 


171396 


70957944 


20.3470 


74530 


1300.6 


134614 


415 


172225 


7*473375 


2o.37 J 5 


74590 


I303.8 


135265 


416 


173056 


71991296 


20.3961 


7.4650 


1306.9 


I359i8 


417 


173889 


725H7I3 


20.4206 


7.4710 


1310.0 


136572 


418 


174724 


73034632 


20.4450 


74770 


I3I3-2 


137228 


419 


I7556I 


73560059 


20.4695 


7.4829 


i3 l6 -3 


137885 


420 


176400 


74088000 


20.4939 


7.4889 


13*9-5 


138544 


421 


177241 


74618461 


20.5183 


7.4948 


1322.6 


139205 


422 


178084 


75151448 


20.5426 


7*5007 


1325.8 


139867 


423 


178929 


75686967 


20.5670 


7.5o67 


1328.9 


14053 1 


424 


179776 


76225024 


20.5913 


7.5126 


1332.0 


141196 


425 


180625 


76765625 


20.6155 


7.5185 


1335.2 


141863 


426 


181476 


77308776 


20.6398 


7.5 2 44 


1338.3 


14253* 


42 7 


182329 


77854483 


20.6640 


7o30 2 


1 341-5 


143201 


428 


183184 


78402752 


20.6882 


7o36i 


1344.6 


143872 


429 


184041 


78953589 


20.7123 


7.5420 


13477 


144545 


43° 


184900 


79507000 


20.7364 


7.5478 


1350.9 


145220 


43i 


185761 


80062991 


20.7605 


7-5537 


i354.o 


145896 


43 2 


186624 


80621568 


20.7846 


7-5595 


1357.2 


146574 


433 


187489 


81182737 


20.8087 


7-5654 


1360.3 


147254 


434 


188356 


81746504 


20.8327 


7-5712 


1363.5 


147934 


435 


189225 


82312875 


20.8567 


7-577o 


1366.6 


148617 


436 


190096 


82881856 


20.8806 


7.5828 


1369.7 


149301 


437 


190969 


83453453 


20.9045 


7.5886 


1372.9 


149987 


438 


191844 


84027672 


20.9284 


7-5944 


1376.0 


150674 


439 


192721 


84604519 


20.9523 


7.6001 


1379.2 


i5!3 6 3 


440 


193600 


85184000 


20.9762 


7.6059 


1382.3 


152053 



208 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 













ClRrr.v. 


No. 


Square 


Cube 


Sq. Root 


Cube Root 






Circum. 


Area 


441 


19448 I 


85766121 


2 I .OOOO 


7.6117 


I385-4 


J 5 2 745 


442 


195364 


86350888 


21.O238 


7.6174 


1388.6 


153439 


443 


196249 


86938307 


2I.O476 


7.6232 


I39I-7 


154134 


444 


197136 


87528384 


2I.O713 


7.6289 


1394.9 


154830 


445 


198025 


88121125 


2I.O950 


7.6346 


1398.O 


155528 


446 


198916 


88716536 


2I.H87 


7.6403 


I40I.2 


156228 


447 


199809 


89314623 


2I.I424 


7.6460 


1404.3 


156930 


448 


200704 


899I539 2 


2I.l66o 


7-65I7 


1407.4 


157633 


449 


201601 


90518849 


2I.1896 


7.6574 


1410.6 


*5%337 


45° 


202500 


91125000 


21.2132 


7.6631 


I4I3-7 


159043 


45i 


203401 


9I73385I 


21.2368 


7.6688 


I416.9 


i5975i 


45 2 


204304 


92345408 


2I.2603 


7.6744 


1420.O 


1 60460 


453 


205209 


92959677 


2I.2838 


7.6801 


I423.I 


161171 


454 


206116 


93576664 


2I.3073 


7.6857 


1426.3 


161883 


455 


207025 


94196375 


2I.3307 


7.6914 


1429.4 


162597 


45 6 


207936 


94818816 


21.3542 


7.6970 


1432.6 


163313 


457 


208849 


95443993 


2I.3776 


7.7026 


1435-7 


164030 


458 


209764 


96071912 


2 1 .4OO9 


7.7082 


1438.9 


164748 


459 


210681 


96702579 


2I.4243 


7-7I38 


1442.0 


165468 


460 


2 1 1600 


97336000 


2I.4476 


7-7 J 94 


I445- 1 


166190 


461 


212521 


97972181 


2I.4709 


7.7250 


1448.3 


1 66^ 14 


462 


2 13444 


98611128 


21.4942 


7.7306 


145 1 -4 


167639 


463 


214369 


99252847 


21.5174 


7.7362 


1454.6 


168365 


464 


215296 


99897344 


2I.5407 


7.7418 


1457-7 


169093 


465 


216225 


100544625 


2I.5639 


7-7473 


1460.8 


169823 


466 


217156 


101 194696 


2I.587O 


7-7529 


1464.0 


170554 


467 


218089 


101847563 


2I.6I02 


7-7584 


1467. 1 


171287 


468 


219024 


102503232 


21.6333 


7-7639 


i47o-3 


172021 


469 


219961 


103161709 


2I.6564 


7-7695 


1473-4 


172757 


470 


220900 


103823000 


2I.6795 


7-775o 


1476.5 


173494 


47i 


'22 1 841 


104487111 


2I.7025 


7-7805 


1479-7 


174234 


472 


222784 


105 1 54048 


2I.7256 


7.7860 


1482.8 


174974 


473 


223729 


105823817 


2I.7486 


7-7915 


1486.0 


i757i6 


474 


224676 


106496424 


2I.7715 


7.7970 


1489. 1 


176460 


475 


225625 


107171875 


21-7945 


7.8025 


1492.3 


177205 


476 


226576 


107850176 


21.8174 


7.8079 


1495-4 


177952 


477 


227529 


Io8 53i333 


21.8403 


7-8x34 


1498.5 


178701 


478 


228484 


109215352 


21.8632 


7.8188 


1501.7 


1 7945 1 


479 


229441 


109902239 


21.8861 


7-8243 


1504.8 


180203 


480 


230400 


1 10592000 


2 1 .9089 


7.8297 


1508.0 


180956 



209 



TABLE III.— SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 

CIRCUMFERENCES AND CIRCULAR AREAS OF NOS. 

FROM i TO 520 (Continued) 



No. 


Square 


Cube 


Sq. Root 


Cube Root 


Circle 


Circum. 


Area 


481 


231361 


111284641 


21.9317 


7.8352 


ISII.I 


181711 


482 


232324 


111980168 


21-9545 


7.8406 


I5I4.3 


182467 


483 


233289 


112678587 


21.9773 


7.8460 


I5I74 


183225 


484 


234256 


1 13379904 


2 2 .OOOO 


7.8514 


1520.5 


183984 


485 


235225 


114084125 


22.0227 


7.8568 


J523.7 


184745 


486 


236196 


114791256 


22.0454 


7.8622 


1526.8 


185508 


487 


237169 


H550I303 


2 2.o68l 


7.8676 


1530.0 


186272 


488 


238144 


116214272 


22.O907 


7.8730 


I533-I 


187038 


489 


239121 


1 1 6930 1 69 


22.1133 


7.8784 


1536.2 


187805 


490 


240100 


1 1 7649000 


22.1359 


7.8837 


15394 


188574 


491 


241081 


118370771 


22.1585 


7.8891 


1542.5 


189345 


492 


242064 


1 19095488 


22.l8ll 


7.8944 


1545-7 


190117 


493 


243049 


119823157 


22.2036 


7.8998 


1548.8 


190890 


494 


244036 


120553784 


22.226l 


7.905I 


I55I-9 


191665 


495 


245025 


121287375 


22.2486 


7.9I05 


I 555-i 


192442 


496 


246016 


122023936 


22.2711 


7.9I58 


1558.2 


193221 


497 


247009 


122763473 


22.2935 


7.92 1 1 


1561.4 


194000 


498 


248004 


T2 35 599 2 


22.3159 


7.9264 


1564.5 


194782 


499 


249001 


124251499 


22.3383 


7.9317 


1567-7 


195565 


500 


250000 


125000000 


22.3607 


7.9370 


1570.8 


196350 


5oi 


251001 


125751501 


22.383O 


7-9423 


1573-9 


197136 


502 


252004 


126506008 


22.4054 


7.9476 


I577.I 


197923 


503 


253009 


127263527 


22.4277 


7-9528 


1580.2 


I987I3 


5°4 


254016 


128024064 


22.4499 


7.958l 


1583-4 


199504 


5o5 


255 25 


128787625 


22.4722 


7-9634 


1586.5 


200296 


506 


256036 


i295542i6 


22.4944 


7.9686 


1589-7 


201090 


507 


257049 


130323843 


22.5167 


7-9739 


1592.8 


201886 


508 


258064 


131096512 


22.5389 


7-9791 


1595-9 


202683 


5°9 


259081 


131872229 


22.56lO 


7-9843 


i599.i 


203482 


5io 


260100 


1 3 265 1 000 


22.5832 


7.9896 


1602.2 


204282 


5ii 


261121 


13343283 1 


22.6053 


7.9948 


1605.4 


205084 


512 


262144 


134217728 


22.6274 


8.0000 


1608.5 


205887 


5i3 


263 1 69 


135005697 


22.6495 


8.0052 


1611.6 


206692 


5i4 


264196 


135796744 


22.6716 


8.0104 


1614.8 


207499 


5i5 


265225 


136590875 


22.6936 


8.0156 


1617.9 


208307 


5i6 


266256 


137388096 


22.7156 


8.0208 


1621.1 


2091 1 7 


5i7 


267289 


138188413 


22.7376 


8.0260 


1624.2 


209928 


5i8 


268324 


1 3899 1 83 2 


22.7596 


8.0311 


1627.3 


210741 


5i9 


269361 


139798359 


22.7816 


8.0363 


1630.5 


2II556 


520 


270400 


140608000 


22.8035 


8.0415 


1633.6 


212372 



APPENDIX 



211 





TABLE IV 


—FACTORS OF 


EVAPORATION 






Temper- 
ature 
of 


Boiler gage pressures in pounds per square inch above 
the atmosphere 


feed- 
water 





5 


10 


15 


20 


25 


30 


35 


40 


45 


° Fahr. 






















32 


1. 187 


1 .192 


1. 195 


1. 199 


I. 201 


1.204 


1 .206 


1 .209 


1 .211 


1. 212 


35 


1. 184 


1. 189 


1 . 192 


1. 196 


I. 198 


1.20T 


1.203 


1 .206 


1.208 


1 .209 


40 


I.T79 


1. 184 


1. 187 


1 .191 


I. 193 


1 .196 


1. 198 


1 .201 


1.203 


1.204 


45 


1. 173 


1. 178 


1. 181 


1. 185 


I. I87 


1 .190 


1 .192 


1. 195 


1. 197 


1. 198 


50 


1. 168 


1. 173 


1. 177 


1. 180 


I. 182 


1. 185 


1. 187 


1 . 190 


1. 192 


1. 193 


55 


1. 163 


1. 168 


1 .171 


1. 175 


I. 177 


1. 180 


1. 182 


1. 185 


1. 187 


1. 188 


6o 


1. 158 


1. 163 


1. 166 


1 . 170 


I .172 


1. 175 


1. 177 


1. 180 


1. 182 


1. 183 


65 


I -153 


1. 158 


1 . 161 


1. 165 


I. 167 


1 .170 


1. 172 


1. 175 


1. 177 


1. 178 


70 


1. 148 


1. 153 


1. 156 


1. 160 


I .162 


1. 165 


1 .167 


1. 170 


1. 172 


1. 173 


75 


1. 143 


1. 148 


1. 151 


1. 155 


I. 157 


1. 160 


1. 162 


1. 165 


1. 167 


1. 168 


8o 


1. 137 


1. 143 


1 .146 


1. 149 


I .151 


1. 154 


1. 156 


'1. 159 


1 . 161 


1. 162 


85 . 


1. 132 


1. 137 


1 . 140 


1. 144 


I .146 


1. 149 


1. 151 


1. 154 


1. 156 


1. 157 


90 


1 .127 


1 .132 


1. 135 


1. 139 


I .141 


1. 144 


1 .146 


1. 149 


1. 151 


1. 152 


95 


1 . 122 


1 .127 


1. 130 


1. 134 


I .136 


1. 139 


1 .141 


1. 144 


1 .146 


1. 147 


100 


1 . 117 


1 . 122 


1 .125 


1 .129 


I. 131 


1. 134 


1. 136 


1. 139 


1 .141 


1 .142 


105 


1 . in 


1 . 117 


1 .120 


1 .123 


I .125 


1. 128 


1 .130 


1. 133 


1. 135 


1. 136 


no 


1 .106 


1 . in 


1 .114 


1. 118 


I .120 


1 .123 


1. 125 


1. 128 


1. 130 


1. 131 


ii5 


1 . 101 


1 .106 


1. 109 


1. 113 


I. 115 


1. 118 


1 . 120 


r .123 


1. 125 


1 .126 


120 


1 .096 


1 . 101 


1 .104 


1. 108 


i . no 


1. 113 


1. 115 


1. 118 


1 .120 


1 .121 


125 


1 .091 


1 .096 


1.099 


1. 103 


1 .105 


1. 108 


1 . no 


1 .113 


1. 115 


1 . 116 


130 


1.085 


1. 091 


1.094 


1.097 


1.099 


1 . 102 


1 .104 


1 .107 


1. 109 


1 .110 


135 


1 .080 


1.085 


1.088 


1 .092 


1.094 


1.097 


1.099 


1 .102 


1 .104 


1. 105 


140 


1.075 


1 .080 


1.083 


1.087 


1 .089 


1 .092 


1.094 


1.097 


1.099 


1 .100 


145 


1 .070 


1.075 


1.078 


1 .082 


1.084 


1.087 


1 .089 


1.092 


1.094 


1.095 


150 


1.065 


1 .070 


1-073 


1.077 


1.079 


1.082 


1 .084 


1.087 


1.089 


1 .090 


155 


1.059 


1.065 


1.068 


1 .071 


1.073 


1.076 


1.078 


1. 081 


1.083 


1 .084 


l60 


1.054 


1.059 


1 .062 


1 .066 


1.068 


1 .071 


1.073 


1 .076 


1.078 


1.079 


165 


1.049 


1.054 


1.057 


1. 061 


1.063 


1.066 


1.068 


1 .071 


1.073 


1.074 


170 


1.044 


1.049 


1.052 


1.056 


1.058 


1. 061 


1.063 


1 .066 


1.068 


1 .069 


175 


1.039 


1.044 


1.047 


1. 051 


1.053 


1.056 


1.058 


1 .061 


1.063 


1 .064 


180 


1.033 


1.039 


1 .042 


1.045 


1.047 


1.050 


1.052 


1.055 


1.057 


1.058 


185 


1.028 


1.033 


1.036 


1 .040 


1.042 


1.045 


1.047 


1.050 


1.052 


1.053 


190 


1.023 


1.028 


1. 03 1 


1.035 


1.037 


1.040 


1 .042 


1.045 


1.047 


1 .048 


195 


1. 018 


1.023 


1.025 


1.030 


1.032 


1.035 


1.037 


1 .040 


1 .042 


1.043 


200 


1 .013 


1. 018 


1. 021 


1.025 


1 .027 


1.030 


1 .032 


1.03s 


1.037 


1.038 


205 


1 .008 


1. 013 


1. 015 


1.020 


1.022 


1.025 


1.027 


1.030 


1.032 


1.033 


210 


1.008 


1.008 


I .Oil 


I. 015 


1. 017 


1 .020 


1.022 


1.025 


1.027 


1.028 


212 


1.002 


1.002 





























15 



212 



ARITHMETIC OF THE STEAM BOILER 



TABLE IV— FACTORS OF EVAPORATION {Continued) 



Temper- 
ature 
of 



Boiler gage pressures in pounds per square inch above 
the atmosphere 



feed- 
water 


50 


60 70 80 


90 100 120 140 


160 ] 


80 


° Fahr. 












32 


1. 214 


1 .217 1 .219 1 .222 


1 . 224 1 . 227 1 .231 1 .234 


1.237 1 


239 


35 


1 .211 


1 .214 1 .216 1 .219 


1 .221 1 .224 1 .228 1 .231 


1.234 1 


236 


40 


1 . 206 


1 . 209 1 . 211 1 . 214 


1. 216 1. 219 1. 2 23 1. 226 


1 .229 1 


231 


45 


1 .200 


1 .203 1 .205 1 .208 


1 .210 1 . 213 1 .217 1 .220 


1.223 1 


225 


50 


1. 195 


1. 198 1 .200 1 .203 


1 . 205 1 . 208 1 .212 1 . 215 


1. 218 1 


220 


55 


1 . 190 


1. 193 1. 195 1. 198 


1.200 1.203 1.207 1. 210 


1. 213 1 


215 


60 


1. 185 


1 . 188 1 . 190 1 . 193 


1. 195 


1 . 198 1 .202 1 .205 


1.208 1 


210 


65 


1. 180 


1. 183 1. 185 T.188 


1 . 190 


1 . 193 1 . 197 1 .200 


1 .203 1 


205 


70 


1. 175 


1 . 178 


1 . 180 1 . 183 


1. 185 


1 . 188 r . 192 1 . 195 


1. 198 1 


200 


75 


1 . 170 


1. 173 


1. 175 1. 178 


1. 180 


1 . 183 1 . 187 1 . 190 


1. 193 1 


195 


80 


1 . 164 


1 . 167 


1 . 169 1 . 172 


1. 174 


1. 177 


1 . 181 1 . 184 


1. 187 1 


189 


85 


1 -159 


1 . 162 


1 . 164 1 . 167 


1 . 169 


1 . 172 


1 . 176 1 . 179 


1. 182 1 


184 


90 


1 -154 


1. 157 


1 . 159 1 . 162 


1 . 164 


1 . 167 


1 . 171 1 • 174 


1. 177 1 


179 


95 


1. 149 


1 • 152 


1. 154 1. 157 


1. 159 


1 . 162 


1 . 166 1 . 169 


1 . 172 1 


174 


100 


1. 144 


1. 147 


1 . 149 1 . 152 


1. 154 


1. 157 


1 . 161 1 . 164 


1. 167 1 


169 


105 


1. 138 


1 . 141 


1 . 143 1 . 146 


1. 148 


1.151 


1. 155 1. 158 


1. 161 1 


163 


no 


1. 133 


1. 136 


1 . 138 1 . 141 


1. 143 


1 .146 


1. 150 1. 153 


1. 156 1 


158 


115 


1. 128 


I. 131 


1. 133 1-136 


1. 138 


1 . 141 


1. 145 1. 148 


1. 151 1 


153 


120 


1. 123 


1 . 126 


1 . 128 1 . 131 


1. 133 


1. 136 


1. 140 1. 143 


1 . 146 1 


148 


125 


1. 118 


1 . 121 


1 . 123 1 . 126 


1. 128 


1. 131 


1. 135 1. 138 


1. 141 1 


143 


130 


1 . 112 


1. US 


1 . 117; 1 . 120 


1 . 122 


1. 125 


1 .129 


1. 132 


1. 135 1 


137 


135 


1 . 107 


1 . no 


I . 112 I . 115 


1. 117 


1 . 120 


1 .124 


1 . 127 


1. 130 1 


132 


140 


1 . 102 


1 .105 


i . 107 i . no 


1 . 112 


1. US 


1 . 119 


1 . 122 


1. 125 I 


127 


145 


1.097 


1 . 100 


1 . 102 


1. 105 


1. 107 


1 .110 


1 . 114 


1 .117 


I . 120 I 


122 


150 


1 .092 


1.095 


1.097 


1 . 100 


1 . 102 


1. 105 


1 .109 1 .112 


1.1.31 


117 


155 


1.086 


1 .089 


1 .091 


1.094 


1 .096 


1.099 


1 . 103 1 . 106 


I . 109 I 


in 


160 


1. 081 


1.084 


1.086 


1 .089 


1 .091 


1.094 


1 .098 1 . 101 


I . 104 I 


106 


165 


1 .076 


1.079 


1. 081 


1 .084 


1.086 


1 .089 


1 .093 1 -096 


1.099 I 


ior 


170 


1 .071 


1.074 


1 .076 


1.079 


1. 081 


1 .084 


1 .088 1 .091 


1.094 1 


096 


175 


1.066 


r .069 


1 .071 


1.074 


r .076 


1.079 


1.083 1.086 


I.O89 I 


091 


180 


1 .060 


1.063 


1.065 


1.068 


1 .070 


1.073 


1.077 


1 .080 


I.O83 I 


085 


185 


1.055 


1.058 


1 .060 


1 .063 


1.065 


1.068 


1 .072 


1.075 


1.078 I 


080 


190 


1.050 


1.053 


1.055 


1.058 


i.o6oji .063 


r .067 


1 .070 


1.073 I 


075 


195 


1.045 


1.048 


1.050 


1-053 


1.055 1-058 


1 .062 


1.065 


1.068 I 


070 


200 


1 .040 


1.043 


1.045 


1.048 


1.050 1.053 1057 


1 .060 


I.063 I 


065 


205 


1.035 


1.038 


1 .040 


1.043 


1.045 1.048 1.052 1.055 


1.058 I 


060 


210 


1 .030 


1033 


1 .035 1 .038 


r . 040 1 . 043 1 . 047 1 . 050 


1-053 1 


055 


212 















APPENDIX 213 

How to Interpolate the Table of Factors of Evaporation 

It sometimes happens when it is desired to use the table of 
factors of evaporation that the given figure for any case falls be- 
tween two certain figures in the table, and therefore the correct 
result cannot at once be found without resorting to what is called 
" interpolation." 

Suppose, for example, that the average steam pressure in the 
boiler is 64 lb. per square inch gage, and that the average tempera- 
ture of the feed water is 13 2 F.; what is the factor of evaporation? 
By referring to the table, there are no columns with heading or side 
heading corresponding to these figures, and unless there is some 
definite method of obtaining exact figures, it would be necessary to 
strike an average between two sets of figures in the table, nearest 
to those given in the example. While for ordinary purposes this 
would be close enough, yet because of the ease with which the 
real figures may be found it is worth while to learn what to do. 

The factor for 60 lb. gage and 130 feed water is 1.115; the 
factor for 70 lb. gage and 130 feed water is 1.118; the factor 
for 64 lb. gage amd 130 feed water is therefore, 

1. 118 — 1. 115 
1. 115+ X4= 1.1162 

In the same manner, the factor of evaporation for 64 lb. gage 
pressure and 140 feed water is found to be 

, 1. 107 — LICK 
1.105 + '— ^X4= 1. 1058 

There is now the factor for 64 lb. gage and 130 feed water, 
and 64 lb. gage and 140 feed water, and it only remains to inter- 
polate between these values to get the factor for 64 lb. gage and 
13 2 feed water. This is done as follows: 

1.1162 — 1.1058 

1.1162 X 2 = 1.1141 

10 

which is the factor of evaporation corresponding to 64 lb. gage 
pressure and 13 2 F. feed water, as given in the example. 

The foregoing method may be applied to any figures within the 
range of the table. 



214 



ARITHMETIC OF THE STEAM BOILER 



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2l6 



ARITHMETIC OF THE STEAM BOILER 



TABLE VI.— KENT'S TABLE OF 
Formula: H. P. =3-33 (A -o.6\/a)\/h. 



Diam- 
eter, 

inches 



Area 
A sq. 
feet 



Effective 
area, 

E=A-o.6\/A, 
sq. feet 



Height of chimney 



50 ft. 



60 ft. 



70 ft. 



80 ft. 



90 ft. 



100 ft. 



Commercial horse-power of boiler 



18 
21 
24 
27 

30 
33 
36 
39 

42 
48 
54 
60 

66 
72 

78 



90 

96 

102 

108 

114 
120 
132 

144 



1.77 
2.41 
3-14 
3-98 

4.91 
5-94 
7-07 
8.30 

9.62 
12.57 
I590 
19.64 

23.76 
28.27 
33-18 
33-48 

44.18 
50.27 
56.75 
63.62 

70.88 

78.54 

95 03 

113. 10 



• 97 
1-47 
2.08 
2.78 

3.58 



48 
47 
57 



7.76 
10.44 
I35I 

16.98 

20.83 
25.08 
29-73 
34-76 

40.19 
46.01 
52.23 
58.83 

65.83 

7322 

89.18 

106.72 



23 
35 
49 
65 

84 



25 
38 
54 
72 

92 
115 
141 



27 
41 
58" 
78 

100 
125 
152 
183 

216 



29 
44 
62 
83 

107 
133 
163 
196 

231 
3ii 



66 



113 


119 


141 


149 


173 


182 


208 


219 


245 


258 


330 


348 


427 


449 


536 


565 



694 
835 



For pounds of coal burned per hour for any given size of chimney, 



APPENDIX 



217 



SIZE OF CHIMNEYS FOR STEAM BOILERS 
(Assuming 1 H.P. = 5 lb. of coal burned per hour) 



no ft. 



Height of chimney 



125 ft. 



150 ft, 



i75ft, 



200 ft, 



225 ft, 



250 ft, 



Commeicial hcrse-power of boiler 



Equivalent 
square 
300 ft. chimney; side 
of square, 

\/E+4 in. 



156 
191 
229 

271 
365 
472 
593 

728 

876 

1038 

1214 



204 
245 



389 
503 
632 

776 

934 

1 1 07 

1294 

1496 
1712 
1944 
2090 



3i6 
426 



551 


595 


692 


748 


849 


918 


1023 


1105 


1212 


1310 


I4I8 


1531 


1639 


1770 


1876 


2027 


2130 


2300 


2399 


2592 


2685 


2900 


2986 


3226 


3637 


3929 


4352 


4701 



981 
1 181 
1400 

1637 
1893 

2167 

2459 
2771 

3100 

3448 

4200 
5026 



1253 
1485 
1736 

2008 
2298 
2609 

2939 

3288 
3657 

4455 
5331 



1565 
1830 

2116 
2423 
2750 
3098 

3466 
3855 
4696 
5618 



2005 

2318 
2654 
3012 
3393 

3797 
4223 
5U4 
6i55 I 



16 
19 
22 
24 

27 
30 
32 
35 

38 

43 
48 
54 

59 
64 
70 
75 

80 
86 
91 
96 

101 
107 
117 
128 



multiply the figures in the table by 5. 



2lS 



ARITHMETIC OF THE STEAM BOILER 



TABLE VII- 


-PROPERTIES OF 


SATURATED STEAMi 


i 


2 


3 


4 


5 


6 


7 


Pressure 
lb. per 
sq. in. 


Temp, 
degrees 


Vol. cu. 
ft. per lb. 


Weight, 
lb. per 
cu. ft. 


Heat of 
the liquid 


Latent 

heat of 

evap. 


Total 
heat of 
steam 


P 


F 


Vor S 


V 


Q 


L or R 


H 


i 


101 .83 


333-0 


0.00300 


69.8 


1034.6 


1104.4 


2 


126. is 


173-5 


0.00576 


94-0 


1021 .0 


1115.0 


3 


141-52 


118. 5 


0.00845 


109.4 


1012.3 


1121 .6 


4 


153.01 


90.5 


.01107 


120.9 


1005.7 


1126.5 


5 


162.28 


73-33 


0.01364 


130. 1 


1000.3 


1130.5 


10 


193-22 


38.38 


0.02606 


161 . 1 


982 .0 


1143.1 


14-7 


212 .00 


26.79 


0.03732 


180.0 


970.4 


1150.4 


20 


228.00 


20.08 


0.04980 


196. 1 


960.0 


1156.2 


25 


240. 10 


16.30 


0.0614 


208.4 


952.0 


1160.4 


30 


250.30 


13-74 


0.0728 


218.8 


945-1 


1163.9 


35 


259-3 


11.89 


0.0841 


227 .9 


938.9 


1166.8 


40 


267.3 


10.49 


0.0953 


236. 1 


933-3 


1169.4 


45 


274-5 


9-39 


0. 1065 


243-4 


928.2 


1171 .6 


50 


281.0 


8.51 


0.1175 


250. r 


923-5 


II73-6 


55 


287.1 


7.78 


0. 1285 


256.3 


9190 


II75-4 


60 


292 .7 


7-17 


0.1394 


262 . 1 


914-9 


1177.0 


65 


298.0 


6.65 


0.1503 


267.5 


911 .0 


1178.5 


70 


302.9 


6.20 


0.1612 


272 .6 


907.2 


1179.8 


75 


307.6 


5-81 


0.1721 


277-4 


903.7 


1x81.x 


80 


312.0 


5-47 


0.1829 


282 .0 


900.3 


1182.3 


85 


316.3 


5.16 


0.1937 


286.3 


897.1 


1183.4 


90 


320.3 


4.89 


0.2044 


290.5 


893.9 


1184.4 


95 


324.1 


4-65 


0.2151 


294-5 


890.9 


1185.4 


100 


327.8 


4-429 


0.2258 


298.3 


888.0 


1186.3 


105 


331.4 


4-230 


0.2365 


302.0 


885.2 


1187.2 


no 


334.8 


4-047 


0.2472 


305.5 


882.5 


1188.0 


115 


338.1 


3.880 


0,2577 


309-0 


879.8 


1188.8 


120 


341.3 


3.726 


0.2683 


312.3 


877.2 


1189.6 


125 


344-4 


3.583 


0.2791 


315.5 


874.7 


1190.3 


130 


347-4 


3-452 


0.2897 


318.6 


872.3 


1191 .0 


135 


350.3 


3.331 


0.3002 


321.7 


869.9 


1191 .6 


140 


353.1 


3.219 


0.3107 


324-6 


876.6 


1192.2 


145 


355.8 


3-H2 


0.3213 


327.4 


865.4 


1192.8 


150 


358.5 


3-012 


0.3320 


330.2 


863.2 


II93-4 


155 


361.0 


2.920 


0.3425 


332.9 


861.0 


1194.0 


160 


363.6 


2.834 


0.3529 


335-6 


858.8 


II94.5 


165 


366.0 


2.753 


0.3633 


338.2 


856.8 


II95-0 


170 


368.5 


2.675 


0.3738 


340.7 


854.7 


II95.4 


175 


370.8 


2.602 


0.3843 


343-2 


852.7 


II95-9 


180 


373- 1 


2.533 


0.3948 


345-6 . 


850.8 


1196.4 


185 


375-4 


2.468 


0.4052 


348.0 


848.8 


1 196. 8 



1 Reproduced by perm 
and diagrams (copyright, 



ission 
1909, 



from Marks and Davis's 
by Longmans, Green & 



steam 
Co.). 



tables 



APPENDIX 



219 



TABLE VII— PROPERTIES OF SATURATED STEAM (Continued) 



I 


2 


3 


4 


5 


6 


7 


Pressure 
lb. per 


Temp, 
degrees 


Vol. cu. 
ft. per lb. 


Weight, 
lb. per 
cu. ft. 


Heat of 
the liquid 


Latent 
heat of 


Total 
heat of 


sq. in. 








evap. 


steam 


P 


F 


Vor S 


V 


Q 


L or R 


H 


190 


377-6 


2.406 


0.4157 


350.4 


846.9 


II97.3 


195 


379-8 


2.346 


0.4262 


352.7 


845.0 


II97.7 


200 


381.9 


2.290 


0.437 


354-9 


843.2 


1198.1 


205 


384.0 


2.237 


o.447 


357.1 


841.4 


1198.5 


210 


386.0 


2.187 


0.457 


358.2 


839.6 


1198.8 


215 


388.0 


2.138 


0.468 


361.4 


837.9 


1199.2 


220 


389.9 


2.091 


0.478 


363.4 


836.2 


1 199 -6 


225 


391.9 


2.046 


0.489 


365.5 


834-4 


II99-9 


230 


393.8 


2.004 


0.499 


367.5 


832.8 


1200.2 


235 


395-6 


1.964 


0.509 


369.4 


831. 1 


1200.6 


240 


397-4 


1.924 


0.520 


371-4 


829.5 


1200.9 


245 


399-3 


1.887 


0.530 


3733 


827.9 


1201 .2 


250 


401 . 1 


1.850 


0.541 


375-2 


826.3 


1201.5 



INDEX 



Allowable pressure, 168 
strain on stays, 161 
Analysis of boiler trial, 72-81 
Angle stiffners for curved sur- 
faces, 148 
Angles, bracing, 187 
Approximate method, areas of 

segments, 52 
Area of diagonal stays, 40 
grate, 62-63 

of head to be braced, 186 
of segments to be braced, 

48, 50, 180 
table of segments, 51 
Areas and circumferences of cir- 
cles, table of, 191 



Boiler problems, 124-125 

trial report, 72-81 

tubes, table of standard, 
214 

water- tube and coil, 154 
Braced segments, 48 
Braces and staybolts, 36, 37 
Bracing, angles, 187 
Brown type of furnace, 143 
Bumped heads, 30, 36, 170 
Bursting pressure in cylinder, 9 
11 

pressure of pipe, 85-86 
Butt straps, single, 19, 20 

joints, 19 



B 



Bars, girder, 53-55 
Board of Supervising Inspectors 
Rules, United States, 
135-156 
Boiler efficiency, 71-72 

feed pipe, size of, 118-119 
heating surface, 60-62 
heads, 30-55 

stiffness of, 1 19-124 
horse-power of, 66-68 
Porcupine, 155 



Cast-iron nozzles, 180 
Chimneys, 11 5-1 18 

table of, 216 
Circles, areas of, table, 191 
Circumference of circles, table 

of, 191 
Coil and water- tube boilers, 154 
Collapsing pressure of fire box, 

95, 96, 97 
of tubes, 65, 66, 126, 127 
cone-shaped flue, 92-93 
Combustion chambers tops, 144 
Commercial efficiency, 71-72 



221 



222 



INDEX 



Concave heads, 30-36, 148 
Cone seam, strength of, 94-95 

-shaped flue, 92-93 
Convex-heads, 30-36, 147 
Corrugated furnaces, 63-64 
Cost of evaporating water, 87- 

88 
Cubes, and cube roots, table of, 

198 
Cylinder, the, 6 

with riveted joints, safe 

pressure of, 28, 29 

D 

Decimal equivalent of fractions, 

197 
Description of riveted joints, 12, 

13 
Design of riveted joints, 14 
Diagonal seam, efficiency of, 
91-92 
stays, 39 
area of, 40 
U. S. rules, 138-139 
Diameter of cylinder, 10 
of sphere, 5 
of stays, 38, 180 
Direct stays, 37-39 
Distance between rows of rivets, 
27, 28 
between stays, 38 
Double butt-strapped, quad- 
ruple riveted joint, 24, 

25 
reinforcing rings, 56-61 
riveted lap joints, 17-18 



Double riveted reinforcing rings, 
56-61 



Efficiency, commercial, 71-72 
of diagonal seam, 91-91 
of grate and boiler, 71-72 
of ligaments, 182 
of riveted joints, 14, 15, 169 
Equivalent evaporation, 68-70 
Evaporation, equivalent, 68-70 
External inspection, 167 
Extracts from Massachusetts 
Rules, 156-190 
from Rules of the United 
States Board of In- 
spectors, 135-156 



Factors of evaporation, table of, 
212 

of safety, 157 
Feed pipe, size of, 11 8-1 19 
Fire box, collapsing pressure of, 

95-96 
Flat heads, 33, 34 

surfaces, 145 
to be stayed, 46 
Formulas for diagonal stays, 41 

for safety valves, 99-114 

for spheres, 5-6 
Fox furnace, 142 

corrugated, 63-64 

U. S. Rules, 140-144 
Fusible plugs, 158 



Girder bars, 53, 55 
Grate area, 62-63 
surface, ratio, 68 

H 



Head, area to be braced, 186 
Heads, boilers, 30-55 

bumped, 30, 36, 179 

concave, 30-36 

convex, 30-36 

flat, ss, 34 

stayed, 36-55 

supported by flange, 45 

thickness of, 34, 35, 46 

unstayed, 30-36 
Heating surface of boilers, 60-62 

ratio, 68 
High joint efficiencies, 21, 22 
Horse-power of boilers, 66-68 
Hydrostatic tests, 168 



INDEX 22: 

Joints, U. S. Rules, 135-138 



Lap joints, double-riveted, 17- 
18 
quadruple-riveted, 18, 19 
single-riveted, 15, 16 
tiiple-riveted, 18,-19 

Leeds furnace, 141 

Ligaments, efficiency of, 182 



M 

Manhole reinforcing rings, 56- 

61 
Maximum pressure, 156 

pressure on shells, 181 
Morrison furnace, 141 

N 

Nozzles, cast-iron, 180 
Number of rivets, 20, 21 



Inspection, annual, 166, 167 
Internal inspection, 166 



Joints, butt, 19 

efficiencies, high, 21, 22 
efficiency of, 14, 15 
lap, 15, 16, 17, 18, 19 
riveted, 12, 13, 14, 15, 16, 
18, 19, 169 



Pipe, bursting pressure of, 85-86 

Pitch of rivets, 26, 27, 125, 126 

of stay bolts in furnaces, 

189 
Plugs, fusible, 158 
Porcupine boilers, 155 
Pressure, bursting, of pipe, 85- 

86 
Properties of saturated steam, 

218 



224 



INDEX 



Purves type of furnace, 142 

Q 

Quadruple-riveted, bouble butt- 
strapped joint, 24, 25 
lap joints, 18-19 

R 

Radius of bumped head, 31, 32 
Ratio of heating to grate sur" 

face, 68 
Reinforcing rings, 56-61 
Report of boiler trial, 72-81 
Rings, manhoJe reinforcing, 56- 

61 
Riveted joints, 12 

efficiency of, 14, 15, 169 
U. S. Rules, 135-138 
Rivets, distance between rows, 
27, 28 
number of, 20, 21 

in reinforcing rings, 59 
pitch of, 26, 27, 125, 126 
securing stays, 44 
shearing strength of, 17 
size of, 26, 27, 160 
in single and double shear, 
21 
Roots, square and cube, 198 
Roper's safety valve rules, 107 
Rows, rivets, number of, 20 
Rules for area of segment, 180 
for diagonal stays, 41, 43 
for spheres, 5-6 
safety values, 99-114 
United States inspectors, 
135-156 



Safe pressure, cast-iron vessels, 
flat cast-iron heads, 
88-89 
in cylinder, 9, 11 
in sphere, 5 
working pressure of boilers, 
127-132 
cylinders with riveted 
joints, 28, 29 
Safety, factors, 157 
valves, 151, 163 
rules, 99-114 
Seam, diagonal, 91-92 
Segments, area of, 48, 180 
to be braced, 48 
table of, 51 
Shearing strength of rivets, 17 
Single butt-straps, 19, 20 

and double shear, rivets 

in, 21 
reinforcing rings, 56-61 
riveted reinforcing rings, 
56-61 
lap joints, 15, 16 
Size of boiler feed pipe, 118-119 
and pitch of rivets, 26, 27 
of rivets, 160 
Solid girder bars, 53-5 5 
Sphere, the, 1 
Split girder bars, 53-55 
Squares, cubes, square roots, and 
cube roots, table of, 198 
Stay bolts, pitch of in furnaces, 

189 
Stayed, flat surfaces, 46 



INDEX 



225 



Stayed head, 36-55 
Stays, diagonal, 39 

U. S. Rules, 138-139 
diameter at bottom of 

thread, 180 
direct, 37-39 

U. S. Rules, 140 
rivets securing, 44 
and stay bolts, 36, 37 
strain on, 161 
Steam table, Marks and Davis', 

218 
Stiffness of boiler heads, 119- 

124 
Strength of cone seam, 94-95 
of cone-shaped flue, 92-93 
of riveted joints, 13 
shearing, of rivets, 17 
Stress in cylinder, 7 

on each inch in the cir- 
cumference of cylinder, 10 
in sphere, 4 
Surface, heating, of boilers, 
60-62 



Tables: I, II, III, IV, V, VI, 

VII, 191-219 
Table: I, Area and circumference 

of circles, 191 

II, Decimal equivalents, 
197 

III, Squares, cubes, square 
roots, and cube roots, 
198 

IV, Factors of evapora- 
tion, 211 



Table: V, Standard boiler tubes, 
214 

VI, Kent's table of chim- 
neys, 216 

VII, Marks and Davis' 
steam tables, 218 

of areas of safety valves, 
152 
of segments, 51 
Tests, hydrostatic, 168 
Thickness of heads, Massa- 
chusetts' Rules, 34 
Ohio Rules, 35 
of plate in cylinders, 10 
in heads, 46 
in sphere, 6 
Tops of combustion chambers, 

144 
Total pressure on shell of 
cylinder, 11 
stress in cylinder, 10 
Triple-riveted lap joints, 18-19 
Tubes, collapsing pressure of, 
65-66, 126, 127 
boiler, standard, 214 

U 

United States Rules, extracts 

from, 135-156 
Unstayed heads, 30-36 

V 

Valves, safety, 99, 114, 163 

W 

Water-tube boilers, r54 
and coil boilers, 154 



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